TWO-SIDED HOMOLOGICAL PROPERTIES OF SPECIAL AND
ONE-RELATOR MONOIDS
ROBERT D. GRAY 1 and BENJAMIN STEINBERG2
March 13, 2026
Abstract. A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-FPn if its group of units is of type FPn. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon’s Identity type theorem for the two-sided homology of one-relator monoids of the form <A |r = 1>. In particular, we show that all such monoids are of type bi-FPo. Moreover, we show that if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most 2, while if r is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid M witli defining relation u = v we show that if there is no nonempty word w such that u,v E A'* w a wA* then M is of type bi-FPcc and the Hochschild cohomological dimension at most 2.
1.		INTRODUCTION
Algorithmic problems in algebra have their origins in logic and topology in the fundamental work of Dehn, Thue and Tictze in the early 1900s; see [7,45,48]. The most important algorithmic question concerning an algebraic structure is the word problem, which asks whether one can decide whether two expressions over generators represent the same element. Markov [49] and Post [53] proved independently that in general the word problem for finitely presented monoids is undecidable. This was later extended to cancellative monoids by Turing [58] and then to groups by Novikov [51] and Boone [8].
Given that the word problem is undecidable in general, a central theme running through the development of geometric and combinatorial group and monoid theory has been to identify and study families all of whose members have solvable word problem; see [9, Chapter 2]. One of the most important and interesting such families is the class of one-relator groups, which were shown to have decidable word problem in classical work of Magnus [46] in the 1930s. The theory of one-relator groups has developed extensively over the last century and continues to be a highly active area of research; see [43] for a recent survey.
One of the challenges in studying one-relator groups is that their geometry can be quite complicated. On the other hand, one-relator groups have turned out to be susceptible for study using topological and homological methods. A classical result of this kind is Lyndon’s Identity Theorem [44] which, interpreted topologically, constructs certain natural classifying
2020 Mathematics Subject Classification. 20M50, 20Μ05, 20J05, 57M07, 20F10, 20F65.
Key words and phrases. Homological finiteness property, bi-FPn, cohomological dimension, Hochschild cohomological dimension, monoid, special monoid, one-relator monoid.
This work was supported by the EPSRC grant EP/V032003/1 ‘Algorithmic, topological and geometric aspects of infinite groups, monoids and inverse semigroups’. The second author was supported by a Simons Foundation Collaboration Grant, award number 849561, the Australian Research Council Grant DP230103184 and Marsden Fund Grant MFP-VUW2411 .
1School of Engineering, Mathematics, and Pliysics, University of East Anglia, Norwich NR4 7TJ, England. Email Robert.D.Gray@uea.ac.uk.
2Dopartment of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USA. Email bsteinberg@ccny.cuny.edu.

Admits a
finite complete
rewriting system

Finitely
presented

Finitely
generated

bi-FPæ

bi-FPn

bi-FP3
(= FHT)

bi-FP2

bi-FP1
(= finitely
dominated)

left- and right-FP3

right-FP

right-FP2

left- and
right-FPi

|left-FP, (= f.g. universal left congruence)

right-FP1
(= f.g. universal
right congruence)

Figure 1. Homological finiteness properties satisfied by monoids that admit finite complete rewriting systems, and all implications between them. The proofs that all these implications hold, and that none of the implications in the diagram are reversible, can be found in the papers [14, 35, 37, 39, 40, 54, 55, 59].
spaces for one-relator groups that can be used to deduce all one-relator groups are IP/, and to determine their cohomological dimension. Some recent important results on one-relator groups that have been impacted by topological and homological methods include: residual finiteness of one-relator groups with torsion [62], the proof that one-relator groups with torsion arc virtually free-by-cyclic [31], and the coherence of one-relator groups [30]. In particular, a key step in the recent breakthrough coherence theorem of Jaikin-Zapirain and Linton [30] was to establish homological coherence of one-relator groups, meaning that every finitely generated subgroup is of type FP2.
This paper is part of a programme of research initiated by the authors in [22-24] aimed at developing topological and homological methods for monoids and then applying these new results and techniques to the class of one-relator monoids. In contrast to one-relator groups, far less is currently known about one-relator monoids, and many fundamental questions about them remain open. Indeed, it is an important longstanding open problem whether the word problem is decidable for one-relator monoids. While in general this problem is open, it has been solved in a number of key cases in the work of Adjan, Adjan and Oganesyan, and Lallement; see [1,2,42].
A natural way to attack the word problem for one-relator monoids is via the theory of string rewriting systems. Indeed, an open problem which is intimately related to the word problem for one-relator monoids is the question of whether every one-relator monoid admits a finite complete rewriting system (meaning a monoid presentation which is confluent and terminating; see [27, Chapter 12]). Since complete rewriting systems give computable normal forms, a positive answer to this question would solve the word problem for one-relator monoids.
Homological and topological methods provide important tools for investigating the question of whether a given monoid admits a presentation by a finite complete rewriting system. The Anick-Groves-Squier theorem, cf. [11], shows that if a monoid admits a finite complete rewriting system, then the monoid must satisfy the homological finiteness properties left- and right-FP. Kobayashi [35] improved this result by showing a monoid admitting a finite complete rewriting is of type bi-FP . There are in fact a range of finiteness properties satisfied by monoids that admit finite complete rewriting systems, and they have received extensive attention in the literature. The homotopical finiteness property FDT (finite derivation type) was introduced by Squier in [56] where it was shown that any monoid admitting a finite complete rewriting system had FDT. The two-sided homological finiteness property bi-FP was introduced in [38], and in [39] it was proved that bi-FP3 is equivalent to the property FHT (finite homological type). Note that Alonso and Hermiller in [3] introduced another homological finiteness property satisfied by monoids admitting finite complete rewriting systems (that they also called bi-FPn) that was later shown by Pride [55] to be equivalent to a monoid satisfying both left-FP7 and right-FPn.
In Figure 1 we have arranged all these properties and the various implications between them. Note that by results in [14, 35, 37, 39, 40, 54, 55, 59] none of the implications in Figure 1 are reversible in general. In particular, in relation to the results we prove in this paper, it is important to stress that it is not true that a monoid satisfying left-FPn and right-FPn is of type bi-FPn.
These homological finiteness properties have also arisen naturally in connection to other areas including: monoids with finitely generated universal left congruence, which is equivalent to left-FPi (see [15, 17,36]), the notion of ancestry in a monoid, due to White from the study of Banach algebras on groups and monoids see [15, Section 3] and [61], and finitely dominated monoids (meaning there is a finite subset that dominates the monoid in the sense of Howie and Isbell [28]), which is equivalent to bi-FP1; sec [36]. Monoids of type left-FP, arc also used to define the higher dimensional BNSR invariants of groups; see [5,6].
Motivated by the open question of whether one-relator monoids admit finite complete rewriting systems, together with the Anick-Groves-Squier theorem, Kobayashi initiated the study of homological finiteness properties of one-relator monoids in the papers [32,34]. In these papers Kobayashi resolved this problem in low dimensions by showing that all one-relator monoids arc FDT and thus arc of type left-FP3 and right-FPz. He then asked the natural question [34, Problem 1] of whether this result could be extended to higher dimension left- and right-FP, for all n. In [22] we gave a positive answer to Kobayashi’s problem by proving that every one-relator monoid is of type left-FPæ and right-FP . Looking at the diagram in Figure 1 it is clear that the remaining main open question in this area is to determine whether all one-relator monoids are of type bi-FP3. Establishing the property bi-FP is the best possible result on homological finiteness properties that one could hope to prove for one-relator monoids since it would imply all other previous results on such properties for this class.
One of the main applications of the general results we prove in this paper is to resolve this problem in the key case of one-relator monoids where the defining relation is of the form r = 1. We obtain results in this paper on one-relator monoids of the form <A |r = 1> by proving more general results for monoids defined by finite presentations of the form (A|w1 = 1,...,ww = 1>. These are called special monoids in the literature. Special monoids were studied in the sixties by Adjan [1] and Makanin [47]. Adjan proved that the group of units of a one-relator special monoid is a one-relator group and solved the word problem for this class of one-relator monoids by reducing the word problem of the monoid to that of the group of units. Makanin proved more generally that the group of units of a k-relator special monoid is a k-relator group and showed that the monoid has decidable word problem if and only if its group of units does. Later Zhang [64] gave a more modern approach to these results via the theory of string rewriting systems.
The main general result in this paper reduces the problem of whether the homological finiteness property bi-FP, holds in a special monoid, to the corresponding question for its group of units. In addition, our methods can be applied to relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. Throughout we shall use cd(G) to denote the cohomological dimension of a group G. Our first main result is:
Theorem A. Let M be a finitely presented monoid of the form
M = (A| W1 = 1,...,we = 1>
and let G be the group of units of M.
(i)		If G is of type FP„ (for 1 < n < 00) then M is of type bi-FPn.
(ii)		Furthermore, the Hochschild cohomological dimension of Μ is bounded below by cd(G) and bounded above by max{2, cd(G)}.
We shall then go on to apply this result to prove the following two-sided Lyndon’s Identity Theorem for one-relator monoids of the form (Α | w = 1>.
Theorem B. Every one-relator monoid of the form (A | r = 1> is of type bi-FP. Moreover, if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most 2, while if r is a proper power then the monoid has infinite Hochschild cohomological dimension.
As explained in Section 8 below. Theorem B gives a positive solution for an important infinite family of base cases towards a proof of the more general conjecture that all one-relator monoids (A | u = v> are of type bi-FPo.
The property bi-FP7 is significantly more robust, and carries more information about a monoid, than either of the properties left-FPn or right-FP,. For instance, for any monoid M if we adjoin a zero element to form M° = M υ {0}, then the monoid M° will automatically be of type left- and right-FP while in contrast for any integer η the monoid MO will be of type bi-FP„ if and only if M itself is of type bi-FP7; see [24, Section 3]. There arc other results that provide evidence that bi-FPn is a more reasonably behaved property than its one-sided counterparts. For instance, in [18] we shall prove that a von Neumann regular monoid with finitely many left and right ideals is of type bi-FPn if and only if all of its maximal subgroups are of type FP7. For left- or right-FP, this result fails to hold since any group with a zero element adjoined is left- and right-FP .
The importance of Hochschild cohomology stems from tire fact that it is the standard notion of cohomology for rings; [26], [60, Chapter 9], or [50]. See [63] for a recent survey article on Hochschild cohomology and its many important uses in algebra. The Hochschild cohomological dimension of a monoid bounds botli its left and right cohomological dimension, but there are examples where all three dimensions are different; see [13.25].
To prove our main results we shall need to introduce and develop an entirely new two-sided analogue of the classical Adjan-Makanin-Zhang theory of special monoids. This makes the results in this paper significantly more difficult than the corresponding one-sided results proved in [24, Section 3]. To give an example of this, implicit in the work of Adjan and Makanin, and explicit in Zhang, is the fundamental result that the submonoid of right units (dually left units) of a special monoid M is isomorphic to the free product of the group of units and a finite rank free monoid. This algebraic description of the right units is key to relating the structure of special monoids and their groups of units. For the two-sided theory developed in this paper the right units R and left units L arc replaced by a single monoid N = {(x,y) εRx L: xy = 1], whicli is a submonoid of M x M°P. Here M°P denotes the opposite of the monoid M (see Section 2 below for the definition of this notion). It turns out, rather miraculously, that for any special monoid M this submonoid N of M x M°P is also isomorphic to a free product of the group of units of M with a finite rank free monoid.
Another key ingredient for studying special monoids is the fact, proved in [24, Corollary 3.12], that any two right Schutzenbeger graphs of a special monoid arc isomorphic to each other (as directed labelled graph) and that modulo these graphs the Cayley graph has a treelike structure. This structural description of the right Cayley graph together witlr results about the left-multiplication action of M on it were then combined in [24, Theorem 3.20] to build a left equivalent classifying space for the monoid. To prove the two-sided results in this paper we need to establish several new results that give a sufficient level of information about the structure of the two-sided Cayley graph of the monoid and the action of M x M°P upon it.
In this way, the results in this paper make new contributions to the theory of special monoids by fully developing, for the first time, the two-sided theory. We note in passing tliat the success of the theory of special monoids has inspired the development of a parallel theory of special inverse monoids initiated in important work of Ivanov, Margolis and Meakin [29] and then developed further in several recent papers including [19-21].
The paper is organised as follows. In Section 2 we give some preliminary definitions and results on presentations, rewriting systems and homological algebra. In Section 3 we give some background on the theory of finitely presented special monoids including results that give normal forms for the elements of the monoid. Section 4 is devoted to proving that the monoid TV, discussed above, has the structure of the free product of a group and a free monoid. In Section 5 we prove that M x M is a free TV-set and apply this in Lemma 5.3 to show that the quotient Γ /TV of the two-sided Cayley graph Γ of M by N is a forest. Section 6 is devoted to the proof that the set E of edges collapsed when forming Γ /TV is a finitely generated free M x M°P-set. In Section 7 we combine all the results from Sections 4-6 together to prove our main results Theorem A and Theorem B. Finally in Section 8 we give a framework for attacking the problem of proving that all one-relator monoids <A|u = v>, explain how Theorem B deals with an infinite family of base cases for that approach, and also sliow how to deal witli the other infinite family of base cases by showing that if a non-special one-relator monoid M has defining relation u = υ such that there is no nonempty word r with u,ve A*r n rA*, then M is of type bi-FPæ and the Hochschild cohomological dimension at most 2.
2.		PRELIMINARIES
Presentations and complete rewriting systems. We recall some basic definitions and notation for monoid presentations and rewriting systems. For further details we refer the reader to [27, Chapter 12]. Let A be a nonempty set, that we call an alphabet, and let A* denote the free monoid of all words over A. Given two words u,v E A* we write u = v to indicate that u and υ arc equal as words in A*, in which case we say the words arc graphically equal. A rewriting system over A is a subset R of A* x A*. The pair (A | R> is called a monoid presentation. The elements of R arc called rewrite rules of the system, and the defining relations of the presentation. We write u —R v for words u,ve A* if there are words a, β E A* and a rewrite rule (e,r) E R such that u = alp and ν = arfl. We use —>1 to denote the reflexive transitive closure of —>R, and write <>1 for the symmetric closure of —>When the set of rewrite rules with respect to which we are working witli is clear from context, we shall often omit the subscript R and simply write —>, ->* and <>*. The relation z is a congruence on the free monoid A* and the quotient A*/<>* is the monoid defined by the presentation (A | R>. When working with a monoid M defined by presentation (A | R>, given two words u,v E A* we write u = v (or sometimes we write u =M v) to mean they are equal in M.
Given a rewriting system R over an alphabet A, we say that a word u E A* is irreducible (or reduced) if no rewrite rule can be applied to it, that is, there is no word υ such that u —> v. The rewriting system R is Noetherian if there is no infinite chain of words Ui E A* with u2 —> Ui+1 for all i > 1. We say that the rewriting system is confluent if whenever u —>* U1 and u —>* u2 there is a word ν E A* such that u1 —>* v and u2 —>* v. A rewriting system that is both Noetherian and confluent is called complete. If R is a complete rewriting system then each <>* equivalence class contains a unique irreducible word, so in this situation the irreducible words give a set of normal forms for the elements of the monoid that is defined by the presentation (A|R).
The category of M-sets. For any monoid M a left M-set consists of a set X and a mapping M X X —> X written (m, x) —> mx called a left action, such that lx = x and m(nx) = (mn)x for all m, η e Μ and x e X. We define right M-scts dually. Note that right M-sets arc the same thing as left M°P-sets, where M°P is the opposite of the monoid Μ which is the monoid with the same underlying set M and multiplication given by x · y = yx. An M-biset is an Μ x M°P-set. We often write (m1,m2)x = m1xm2 in this case. A mapping f: X —> Y between M-sets is said to be M-equivariant if f(mx) = mf(x) for all x e X, m e M. For a fixed monoid Μ the collection of M-sets together witli M-equivariant mappings form a category.
A left M-set X is a free left M-set on B if there is a subset B c Χ such that each element of X can be expressed uniquely as mb witli m e Μ and be B. In this situation we call B a basis for X. Free left M-sets have the usual universal property. The free left M-set on B exists and can be realised as the set Μ x B witli action m(m‘,b) = (mm‘,b). Free right M-sets are defined dually. If X is a free left M-set witli basis B, then ZX is a free left ZM-module with basis B.
Given any right M-set X we define a natural preorder relation < on X where x < y if and only if xM g yM. We write χ & y if there is a sequence 21,22,...,En of elements of X sucli that for eacli 0 < i < η — 1 either z2 < %i+1 or z2 2 Zi+1. This is an equivalence relation and we call the æ-classes of X the weak orbits of the M-set. This corresponds to the notion of the weakly connected components in a directed graph. If X is a right M-set then we use X/M to denote the set of weak orbits of the M-set.
Homological algebra. Let R be a ring. Then a left R-module V is said to be of type FP7 if it has a projective resolution P. —> V —> 0 sucli that Pi is finitely generated for 0<i< n. This is equivalent to having a free resolution that is finitely generated through degree n; see [12, Proposition 4.3]. We say that V is of type FPæ if it has a projective (equivalently, free) resolution that is finitely generated in all degrees. A right R-module is FP7 if it is FP7 as a left R°P-module, and we say an R-R-bimodule (called here an R-bimodule) is FP7 if it is FPn as a left Re = REz R^-module. Note that if M is a monoid, then ZM° = Z[M χ M°P]. We also write FPx(R) to indicate that we arc referring to the ring R in the case that ambiguity might arise. For convenience, all modules will be considered FP-1. A ring R is of type bi-FPn if the R-bimodule R is FPx(Re). We say that a left R-module V has projective dimension at most d if it has a projective resolution of length d.
A monoid M is of type left-FP„ (respectively, right-FPn) if the trivial left (respectively, right) ZM-module Z is of type FPx. Kobayashi and Otto [38] defined Μ to be of type bi-FPn if ZM is of type bi-FP7. So, the monoid M is of type bi-FPn if there is a projective resolution
■ ■ ■ Pi Pi ZM ^()
of the ZM-bimodule ZM, where P0, Pi,..., Pn are finitely generated projective ZM-bimodules. If a monoid is of type bi-FP.„ then it is botli of type left-FP, and right-FP,; see [39, Proposition 4.2].
Kobayashi [35] proved that if a monoid admits a presentation by a finite complete rewriting system then that monoid must be of type bi-FPo. See also [23, Section 11] for a topological proof of this using the theory of equivariant collapsing schemes (i.e., discrete Morse theory).
The left (right) cohomological dimension of a monoid M is the projective dimension of the trivial left (right) ZM-module Z. For a group G the right cohomological dimension is equal to the left cohomological dimension and is just called the cohomological dimension of the group and is denoted cd(G). The Hochschild cohomological dimension of a monoid M, written dim M, is the projective dimension of ZM as a Z[M χ M^-modulc. The Hochschild cohomological dimension bounds both the left and right cohomological dimension of the monoid.
We shall need the following result on modules of type FP„which may be found in [4, Proposition 1.4].
Lemma 2.1. Let R be a ring. If 0 —> A —> B —> C —> 0 is an exact sequence of left R-modules such that A is of type FPn-1 and B is of type FPn, then C is of type FP7.
We shall also make use of following result about projective dimension of R-modules. The following lemma can be proved by applying [10, Lemma 1.5]; see [24, Corollary 2.4].
Lemma 2.2. Let R be a ring and suppose that
Ch —Cu-l — 		Co— V-0
is a partial resolution of an R-module V. Let d 2 η and suppose that Cn —> Cn-1 is injective. If Ci has a projective dimension of at most d—i, for 0 <i< η, then V has a projective dimension at most d.
In [24, Section 3] a key part of proving results relating tire one-sided homological finiteness properties of a special monoid and those of its group of units involved analysing properties of the right and left Cayley graphs of the monoid. In this paper, to study bi-FP„ and the Hochschild cohomological dimension we shall need to prove results about the structure of the two-sided Cayley graph of the monoid M which is defined as follows. Let Μ be a monoid generated by a finite set A. Then the two-sided Cayley graph T = T (M, A) of Μ with respect to A is the directed labelled grapli witli vertex set M x Μ and directed edges M x A x M where (m1,a, m2) is a directed edge with initial vertex (m1, am2) and terminal vertex (mia, m2). Just as a monoid acts naturally by left multiplication 011 its right Cayley graph, there is a natural action of M χ M°P on Γ = Γ (M, A) via
(m2, m2) · (x, y) = (m1x,ym2)
(m1,m2) ■ (x,a, y) = (m1x,a, ym2)
where (m1,m2) € M x M°P, x,y e Μ and a e A. These are free M x M°P-sets witli bases
{(1, 1)} and {1} x Ax {1}, respectively. The two-sided Cayley grapli and this action of Μ x M°P
will play a central role in the proof of the mains result of this paper. See [23, Section 5.3] for
more on the two-sided Cayley grapli and its role in the study of two-sided homological and
geometric finiteness properties of monoids.
3.		FINITELY PRESENTED SPECIAL MONOIDS
We now give some definitions and results due to Zhang [64] that will be used throughout the paper. Let M = ZA | W1 = 1,...,w = 1> and let G be the group of units of M. By [64, Theorem 3.7], G has a group presentation witli k defining relations. We call a word u € A* invertible if it represents an element of the group of units G. We say that an invertible word u e A* is indecomposable if it has no proper prefix that is invertible. It is straightforward to sec that every nonempty invertible word u has a unique factorisation u = U1...u1 where eacli uj is nonempty and indecomposable. For each relation word w; in the presentation of M let w; = w;I...ir,,, be its decomposition into nonempty indecomposable invertible words. Given any invertible word β e At we call β minimal if it is indecomposable, and in addition |B| < maxisisk w;. Now let Δ be the set of all minimal invertible words 8e A* sucli that 8 = Wij in M for some 1<i<n,1<j< nj. It is immediate from the definition that Δ is a finite set of indecomposable invertible words. It is proved in [64, Section 3] that the elements represented by words from Δ give a finite generating set for the group of units G of M.
An important result of Zhang [64] shows how Δ can be used to give an infinite rewriting system for the monoid M modulo the group of units of M, in the following way. Let <s be the sliort-lex order on A* induced by some fixed order on the generators A. Then define a rewriting system
S = {u —> υ | u,v e A*,u = v in M and u >s v}.
It follows from [64, Proposition 3.2] that S is an infinite complete rewriting system defining the special monoid M. We call a word a € A* irreducible (or reduced) if it is irreducible with respect to the rewriting system S. For any word we A we shall use w to denote the unique reduced word sucli that w = w in M.
The following lemma is folklore.
Lemma 3.1. Let T be a monoid and r,s,t e T with rs and st invertible. Then s and rst are invertible.
Proof. First note that rst[(st) 1s(rs) 1] = 1 and [(st) 1s(rs) 1]rst = 1, whence rst is invertible. Since rs, st are invertible, s is left and right invertible, hence invertible.	□
A key result for us is the following.
Lemma 3.2. [64, Lemma 3.4] An irreducible word u E A* is invertible if and only if u Ε Δ*.
Another result that will prove useful later on is the following normal form result of Otto and Zhang [52, Theorem 5.2 and Lemma 5.4]. We shall say a word we A* is trivial if it represents the trivial element of M. We say a subword u of a word w is a maximal invertible subword if u is an invertible word and u is not properly contained in some other invertible subword of w. Lemma 3.1 implies that any nonempty invertible subword u of a word w is contained in a unique maximal invertible subword of w.
Lemma 3.3. [52, Theorem 5.2] Let M = <A | wi = 1,..., W = 1>. Then every word u E A* can be uniquely factorised as u = U0@1U1...amUm where a2 Ε A, for 1<i<m, and Uj is a maximal invertible subword of u for 0 < j < m. We call this the Otto-Zhang normal form of u. Moreover, if ν E A* with u = ν in M, then the Otto-Zhang normal form of υ is ν = ν$αγθγ... amvm where u^ = Vi in Μ for i = 0,...,m.
We note that some of the u^ and vj in the lemma above may be empty.
Lemma 3.4. Let M = (A| wi = 1,..., ww = 1 >, let u € A* and let
u — u0a1u1 · · ■ amum
be the Otto-Zhang normal form of u. Then u = U0a1U1---@mUm and each u; E Δ*. Moreover, this is the Otto-Zhang normal form of u.
Proof. To see this, it suffices to show that each u: is a maximal invertible subword of v == U0a1U1---amUm, and that v is reduced. Suppose that w is a maximal invertible subword of v. Since each Uj is invertible, it follows from Lemma 3.1 that if w contains a letter from some Uj then it must contain the entire uj. It follows that either either w = ffi for some i, or w = U1-1aL---a*Uk for some 1 < i < k < m. But in the latter case U0—102---aUK is an invertible word, a contradiction. Thus each uj is a maximal invertible subword of υ. It now follows that v is reduced since the left-hand side of each rewrite rule is nonempty and invertible, and each maximal invertible subword of υ is reduced. Finally, by Lemma 3.2, each u2 is a word in Δ*.□
Let us say that a word u E A*, written in Otto-Zhang normal form
u = u0a1u1 . . . amm
has no invertible suffix if Um represents the identity of M, that is Um is empty. We define having no invertible prefix dually. In particular, a nonempty reduced word μ has no invertible suffix if the Otto-Zhang normal form of μ is μ = μ^α^μγ ■ ■ ■ μηαη witli the ai Ε A and the pi the maximal invertible subwords of μ. In particular, μ is not invertible. There is an obvious dual condition for any nonempty reduced word to have no invertible prefix.
4.		Proving THAT N =G *C* FOR SOME FINITE SET C.
Throughout this section
M = (A| = 1,...,w = 1>.
Let R1 and L1 be the right and left units, respectively, of the monoid M. Let G = R1 o Li be the group of units of M. Define a monoid
Ν = {(u, v) : u E R1,v E L1,uv = 1}
with multiplication
(a,b) ■ (x,y) = (ax, yb).
Note that N is a submonoid of R1 χ LQD gM x M°P.
We remark that while N is a set of pairs of elements of Μ we shall almost always write elements of TV as pairs (u, υ) where u, υ e A* . This should of course be interpreted as meaning the pair ([u], [p]) where [u], [w] ε M arc the elements of M represented by the words u and υ, respectively.
By results of Zhang [64, Theorem 4.4] we know that R1 = G * C* for some finite set C, and Li ^G * D* for some finite set D, it follows that TV is a submonoid of (G * C*) χ (G * D*)°P. This implies that N is group-embeddable, and thus, in particular, N is two-sided cancellative. More generally R1 x L^ is a group-embeddable and thus two-sided cancellative monoid.
Our aim in this section is to prove that N ^G * X* for some finite set X. The key to proving this result is to identify a suitable generating set with respect to which we can see that TV has this free product structure.
Given any invertible word ο e A* we shall use α-1 to denote the reduced form of the inverse of tlie invertible element a. For example if Μ = (A\r = 1) where r = (ab)(cd)(ab) then (cd)(ab) = (ab)(cd) is the inverse of ab and it may be verified that (ab)^1 = (ab)(cd) is the normal form in this case. By Lemma 3.2 we know that a-1 e Δ* for every invertible word a e A*, and if a is itself irreducible, then also a e Δ*.
Let u, v e A+ be reduced words sucli that uv = 1 in M. Then by definition of the Zhang rewriting system and the fact that u, ν are reduced, there must exist factorisations u = u\U2 and ν = ν\ν2 witli uzvi e Δ* and 112 and νι both nonempty. Of course, U2;V\ is an invertible word. In general, if u,v are nonempty words, we say that an invertible word ze A+ cuts across (u,v) e A+ x A+ if u = uyU2 and ν = V1V2 with 2 = u2vi and U2,U1 nonempty. Let (u,v) e N witli u and v botli reduced. It follows from the definitions that u = 1 in Μ if and only if υ = 1 in M. Otherwise, u e A+ and v e A+ and the argument above shows that at least one invertible word cuts across (u, υ). The collection of invertible words cutting across (u,v) is partially ordered by subword inclusion, as subwords of uv.
Here a subword of uv cutting across (u,v) is specified by choosing a letter χ in u and a letter y in v and taking the unique subword of uv that begins witli that particular letter x and ends with that particular letter y. Distinct pairs of such letters x, y determine distinct subwords of uv, viewed as subwords of uv, even if graphically they arc the same word. For example, suppose M = (a,b | (ub)® = 1). Let a = aba and v = bab so that uv = ababab. Then this word uv contains the word abab as a subword in two different ways, once as a prefix ababab of uv and once as a suffix ababab of uv. Eacli of these subwords abab of uv cuts across (u,v) = (aba, bab) but, even though they are both equal to the word abab, they are distinct as subwords of uv, and thus they are two distinct elements in the poset of invertible words cutting across (u,v). Furthermore since as subwords of uv neither of these words abab is contained in the other, they are incomparable in this poset. Note that the subword ab cutting across uv, which is the underlined word in ababab, is also an element of the poset of invertible words cutting across (u,v), and this element of the poset is contained, as a subword of uv, in both the subwords abab of uv and hence lies below eacli of those elements of the poset.
Returning to the general argument, we now aim to show that set of invertible words cutting across (u,ν), if nonempty, is a lattice. Let us prove a more general result first.
Proposition 4.1. Let u e A+ and v be a nonempty subword of u. If some invertible subword ofu contains v, then the set of invertible subwords of u containing ν is a lattice ordered by subword inclusion. Moreover, if δ = U\VU2 is the unique smallest element of this lattice, then ui has no nonempty invertible prefix and u2 has no nonempty invertible suffix.
Proof. We being by showing the poset is a lattice. Let x, y e A+ be invertible subwords of u containing v. We must show that x,y have a join and meet in the collection of invertible subwords of ν containing u. Write x = xivx2 and y = yivy2· Without loss of generality assume that Iii| <|yi|. If |ΐ2 ^|?/2|, then χ is a subword of y and so the meet and join in this case arc clearly given by x and y, respectively. So assume that [λ] > |ϊ/2|· Then it suffices to show that yivx2 and xivy2 arc invertible, as these will then be the meet and join, respectively, of x,y in this poset. Write yi = ax1 and x2 = y2B. Then Y1vy2 = xx1vy2 and x1vx2 = x1vy23 arc invertible, and so x1vy2 and Y1vx2 = xx1vy2B arc invertible by Lemma 3.1, as required.
For the final statement, if ui = u/u with u1 invertible, then ufvu2 is invertible and hence by minimality of u, we must have u1 = U1, and so u1 is empty. Dually u2 has no nonempty invertible suffix.	□
Proposition 4.2. Let (u,v) e 4+ χ At. Suppose that some invertible word cuts across (u,v). Then the poset of invertible words cutting across (u,v) is a lattice.
Proof. Let a be the last letter of u and b the first letter of υ and consider the subword ab of the word uv where ab cuts across (u, v). Then the invertible words cutting across (u, v) are the subwords of uv containing this particular subword ab of uv. Therefore, this poset is a lattice by Proposition 4.1.	□
Definition 4.3. Given any pair (p,?) e At χ At such that some invertible word cuts across (p,?), we define the height of (p,Y) to be the height of the lattice of invertible words cutting across (p,?), i.e., the length of the longest chain of invertible words cutting across (p,).
Clearly since the lattice in Definition 4.3 is always finite, it must have finite height, and so the height of a pair of words (u,?) € At x At, such that some invertible word cuts across (u, 7), is always finite.
We shall later need the following property of height 0 pairs.
Lemma 4.4. Let u,?Y e A* be reduced with py invertible but μ noninvertible (or, equivalently, Y noninvertible). If (u,Y) has height 0, then μ has no nonempty left invertible prefix and 7 has no nonempty right invertible suffix.
Proof. Write μ = μ'α and 7 ξ b^' with a,b e A. Note that μ' is right invertible and Y is left invertible, and so μ' has no nonempty left invertible prefix and Y has no nonempty right invertible suffix by Proposition 4.1. Since μ and 7 arc also not invertible, but arc right and left invertible, respectively, the lemma follows.	□
We are now ready to provide a finite generating set for N.
Lemma 4.5. The monoid N is generated by the set X OY where
Υ = {(6,6-1) : e A}
and X consists of all pairs (81,828-1) such that:
(1)		81,02 € At are reduced;
(2)		δ = δ±δ2 is invertible;
(3)		(61,62) has height 0;
(4)		81 has no invertible suffix.
Furthermore these four conditions imply that δ = δγδ2 e Δ, and hence the set X is finite. Therefore N is finitely generated by the set X oY.
Proof. Let u,v € A* be reduced with uv = 1 in M. We prove by induction on u that (u,v) € (X UY). If u = 1, then since uv = 1 in M and v is reduced, we must have v = 1. Proceeding by induction, if u > 0, then uv is not reduced (being 1 in M), and so wc can apply a rewrite rule to it. Since u, ν arc reduced, the left-hand side of any applicable rewrite rule cuts across (u,v). But the left-hand side of any rewrite rule is invertible and nonempty. Hence there is an invertible word cutting across (u,v). Let δ be the minimal invertible word cutting across (u,v); this exists by Proposition 4.2. Write u = u‘ò1 and v = d2v‘ with δ = δγδ2- Suppose first that 81 has no invertible suffix. We compute (u,v) = (u‘, ov‘)(81, 828 1). Note that u‘< u and u'δν' = uv, and hence is 1 in M. So (u‘, δν') e <X UY) by induction, noting that u‘ is a reduced word, whereas (81, 826-1) € X by minimality of 8 which implies (81, 82) has height zero. Otherwise, 81 = By where 7 is a nonempty invertible suffix. Since 81 is reduced, 7 e A* by Lemma 3.2. Therefore, (7,7-1) ε (F)· But then (u,v) = (u’ofy,v) = (u‘81,70)(4,71), where u’dl is a reduced word since it is a subword of the reduced word u. Since u’òlyv = uv = 1 in M, and u‘d{ < u, we deduce that (u’oly, v) e <X U Y> by induction, and hence (u, v) e <X υ Y>.
Finally, we show that conditions (1)-(4) together imply that 8 = 0102 e Δ and hence the set X is finite. There are two cases to consider.
First suppose that 5 = 0102 is not reduced. Since 81 and 82 are both reduced by (1) it follows that the left hand side μ e At of some rewrite rule cuts across (01, 82). Write 8102 = 81u1u205 where 81 = 81u1, 82 = /205 and /1/2 e At with /1 and p2 botli nonempty words. Since by (4) the word 81 has no invertible suffix, it follows that /1 4 At hence there is a word 7 e Δ sucli that 7 cuts across (81,82). Since by (3) we have that (81,82) has height zero this is only possible if δ = 5152 s7eΔ. This completes the proof that in this case 5 = 5152 € Δ.
Next we consider the case that 5 = 5152 is reduced. Since 8182 is reduced, then as it is invertible it belongs to Δ* by Lemma 3.2. Since 5ι 4 Δ* (being noninvertible), the expression of 5152 as a product of words from Δ must involve an element 57 e Δ cutting across (81,02). Since any element of Δ is invertible, we deduce that 5152 = 5z e Δ by definition of height 0. This completes the proof that in this case 5 = 5152 € Δ.
We have shown that the conditions (l)-(4) together imply that 5 = 0102 e Δ and hence, since Δ is finite, the set X is also finite. □
Remark 4.6. The idea behind the height zero condition in the definition of the set X in Lemma 4.5 is to ensure that the generating set obtained in minimal. In particular, if we were to remove the height zero assumption from the set X in Lemma 4.5 then in general it would result in a generating set for N that is not minimal. To see this consider, for example, the monoid M defined by the presentation <a,b,c, d | (abcd)(bc)(abcd) = 1>. It is straightforward to verify that the minimal invertible pieces of this defining relator are abed and bc. Consider the following three elements from N
(b,c(bc)~1), (ab, cd(abed)~1), (a,bcd(abed)~1).
All three of theses elements has the form (5ι, 820-1) where 5 = 8182 e Δ since it may be shown that be e A and abed € Δ. However, if we included all three of these generators in a generating set for N then that generating set would not be minimal since
(a, bcd(abcd)-1) ■ (b, c(bc)-1) = (ab, c(bc)~1bcd(abed)~1) = (ab, cd(abcd}~1{.
Note that the pair (ab, cd(abcd)~1) docs not have height zero, and the product above shows that this element can be written as a product of the two height zero elements (a, bcd(abcd)~1) and (b, c(bc)~1).
We are now in a position to prove that N decomposes as a free product of its group of units and a free monoid of finite rank. Let us recall briefly the normal form for a free product. Suppose that [Mi : i e I} is a collection of monoids. Consider the alphabet B = Liter M\{1}. An element of M\{1} is called an M2-syllable or a syllable of type Mi. A word in B* is in normal form if it does not contain two consecutive syllables of the same type. The lengtli of a normal form is called its syllabic length. Eacli element of the free product *ieIM has a unique representative in normal form.
The following is a variant of the Ping Pong Lemma from group theory (see e.g. [16, Chapter II.B] or [45, Proposition 12.2 of Chapter III]).
Lemma 4.7 (Ping Pong Lemma). Let Ν be a cancellative monoid, G a subgroup of the group of units of N and XgN. Suppose that N = (G oX> and there is a right N-set A such that A = A0 uLex Ax with Axg — A0 whenever x e X and 1# g € G, and Ax $A. Then <X > is a free monoid with basis X and N = G* X*.
Proof. First note that if X = Ø, then Ν = G and there is nothing to prove. So we may assume that X = 0. For each x e X if xl = x3 for some i, j e Ν witli i > j then since N is a cancellative monoid this would imply xt-j = 1 and thus x is invertible. But if x were invertible it would imply Ax = A contradicting the hypotheses of the lemma. Hence, eacli x € X generates a free submonoid x*. We want to show that N = G* **EXx*. By assumption, the natural homomorphism φ : G * **Xx* —> N from the free product to N is surjective. The action of N on A can be used to define an action of G * **xx* on A via the homomorphism φ. More precisely, we define an action of G * **Xx* on A where for each seG* **Xx* and ae A we define as = ad(s). Next note that if an element « of the free product G * **6Xx* is in normal form with last syllable of type x*, then Aog Ax. If a has last syllable of type G, there are two cases. If a € G\{1}, then Aa = A and Αχα § A0 for all x e X; if a = a'xkg with x e X, k >0 and g E G\{1}, then Aa • A0. It follows that if a is a nonempty normal form, then it acts nontrivially on A, and we can determine the type of its last syllable from its action on A.
Suppose that Q and β are normal forms for the free product. We induct on the sum of their syllable lengths to show that if a and β represent the same element of N, then they are equal in the free product. By the previous paragraph, we may assume that a and β are nonempty and their last syllables have the same type. Suppose first that a = a'xk and β = β'χ™ with χEX and a'.β' of one shorter syllable length. Without loss of generality assume that k< m. Then cancelling xk we get a' and rxm-k represent the same element of N. By induction, since a' does not end in an x*-syllable, we must have that m = k and a' = β' in the free product. Next suppose that a = a'g and β = β'g' where g, g' e G\{1} and αββ' have syllable length one shorter. Then d‘ = B‘g‘g-1 in N. Since d‘ does not end in a G-syllable, we deduce by induction that g’g 1 = 1 and d‘ = β' in the free product. This completes the proof. □
We now apply the Ping Pong Lemma to our setting.
Theorem 4.8. The monoid Ν is isomorphic to a free product G * C* where G is the group of units of M and C is a finite set. More precisely, let X v Y be the finite generating set for Ν obtained in Lemma 4.5. Then X is a basis for a free submonoid of Ν and N = G * X* where the copy of G is identified with (Y).
Proof. Note that if μ, y € A* arc reduced with μ nonempty, and with py invertible and with μ having no invertible suffix, then μ is right invertible, but not invertible.
We prove the free product decomposition using the Ping Pong Lemma applied to the right action of N on itself. We know that N is generated by X o Y by Lemma 4.5. We have already pointed out that N is cancellative. Note that if x = (81,020 1) θ X with 8102 = δ E A and 81 having no invertible suffix, then 01 is right invertible (as 5 is invertible), and hence is not left invertible. It follows that χ is not left invertible, and so Nx C N for all x e X. We now prove that the collection {Nx : χ e X} is pairwise disjoint.
Let me N with π = (πρ, πβ) where πρ and xr are reduced words. If (p, YT-1) € X with pY = Τ € Δ. we shall show that we can uniquely recover (u, YT 1) from (TOP, YT 1,) = T(p, YT 1). We need a claim in order to do this. We continue to write w for the reduced form of w.
Claim 1. Let π = (ττ^,πΓ) e Ν with πρ, πΓ reduced, (μ, ^τ χ)e X with μμ = τΕ Δ and δΕ Δ* reduced. Then
π(μ,	5-1) = βπ^μδ,
and χ^μδ and μτ Xr are both reduced.
Proof of Claim 1. By assumption Tg, xr and δ are reduced. The words p, are reduced by the definition of X in Lemma 4.5. First we observe that Tep is reduced. This follows since Tg and μ arc reduced, and no rewrite rule can be applied with the left-hand side of the rewrite rule including both letters in Tg and in μ since the left-hand side of every rewrite rule belongs to Δ+, and hence is invertible, and that would then give a nonempty left invertible prefix of μ, something we already observed in Lemma 4.4 it does not have. Dually, YT-17, is reduced because no rewrite rule can be applied to Y or T 17, since they are reduced, and furthermore, any rewrite rule applied across both of these words (i.c., involving letters from both of them) would contradict the fact (proved in Lemma 4.4) that Y has no nonempty right invertible suffix, as the left-hand side of any rewrite rule is nonempty and invertible.
Next we prove that δ is a maximal invertible subword of Tepd. Indeed, by definition of the generating set X the word μ does not have an invertible suffix. It follows that there is no suffix μ" of μ such that μ"δ is invertible. Also, a suffix of π^μδ of the form σμδ cannot be invertible since if it were, then σμ would be invertible, implying μ is left invertible. But since by definition μ is right invertible this would imply μ is invertible, and this would contradict the fact that μ has no invertible suffix. Hence J is a maximal invertible subword of Tepò.
Finally we show that π^μδ is reduced. Since πρμ and δ are botlr reduced the left hand side of any rewrite rule applied to π^μδ must cut across (π^μ,δ). But tlie left hand side of any rewrite rule is nonempty and invertible so there would be an invertible word cutting across (πιμ,δ). But then since δ is invertible, by Lemma 3.1 that would imply that 8 is not a maximal invertible subword of Tepò, contradicting what was proved in the previous paragraph. Hence Teuò is reduced. This completes the proof of the claim.	□
From Claim 1 it follows that Tep and YT-17, arc reduced. We now recover (u, 77-1) uniquely from (Tg, YT-17,) in the following way. Consider all the invertible words cutting across this pair. There is at least one such word since pY is invertible and μ and Y are both nonempty words by definition of X. By Proposition 4.2 the set of invertible words cutting across this pair is a lattice under inclusion. We claim that the word PY is the unique minimal element of this lattice. This word is minimal because, by definition of X, the pair (u,3) has height zero. In a lattice minimal elements arc unique. It follows that we can recover (u,YT-1) from (TCP, YT-17,), and hence the collection {Nx : x € X} is pairwise disjoint.
Let No be the complement of Lirex Nx. By the Ping Pong Lemma, it remains to show that Nx(g,g 1) g No for all χ e X and g# 1 in G. Indeed, let χ = (p,T 1) e X, and let δe A* be reduced representing g, whence δ is nonempty. Let π = {τΐ^ 7,) e N witli TTg, Tr reduced. Then mx(g,g 1) = (Teuò, 8-lyr-17,) and Tepo is reduced by Claim 1. Now if ^xig-g *) = n‘x‘ with 7’ = (72, 74) e N, x‘ = (u‘,‘(-‘)-1) e X, t‘ = p’y € A and TTg, 74. reduced, then we would have n‘x‘ = (2u‘, ‘(-‘) 114) witli T2‘ reduced by Claim 1. Thus T2‘ = τ^μδ. We cannot have 8 as a suffix of μ' since μ' has no invertible suffix and δ is nonempty. If p‘ is a suffix of δ, then it is left invertible, but it is also right invertible since p‘<‘ is invertible. This contradicts that p‘ has no invertible suffix. Thus mx(g,g-1) e N0. This completes the proof that X freely generates X* and Χ = G*X* by the Ping Pong Lemma.	□
5.		PROVING M X M IS A FREE N-SET
As in the previous section, throughout this section
M = (A | W1 = 1,...,w = 1>.
For the next part of the argument we want to prove that M x Μ is a free right N-set under the action
(m1,m2) - (n1,n2) = (m1n1,n2m2),
where (m1, m2) € Μ x Μ and (n1,n2) € N. Note that in the above situation we have
m1m2 = m1n1n2m2
in Μ since nin2 = 1 by definition of X.
We shall need the following remark. If S' is a cancellative monoid and X is a free right S-set, then xs = xs‘ implies s = s’ for x e X and s, s' e S, and dually for free left S-sets. Indeed, this is true for the right action of S on itself by cancellativity, and a free S-set is a disjoint union of copies of the action of S on itself.
Our aim in this section is to prove the following result. Note the basis in the following theorem is not finite, but we do not expect nor need it to be so.
Theorem 5.1. The set M X Μ is a free right N-set under the action
(m1,m2) ■ (n1,n2) = (m1n1,n2m2),
where (m1, m2) e Μχ Μ and (n1,n2) € Ν. In more detail, let B be the set of all pairs of words (01, α2) such that 3132 = a is a reduced word and a has Otto-Zhang normal form
O' — W0@1w1 · · ·dmwm
with 01 = W001---W02—1 and α2 = Wi-101---@mWm for some 1<i< m+1. Then M X M is a free right N-set with basis B.
Proof. Let u,v € A* be reduced words and let uv = W0d1---@mWm in Otto-Zhang normal form. Suppose, moreover, that u = W000 ••• W1-201-1W4 1 and ν = w'^ai ■ ■ - amwm where Wi-1 = ww, 1. Note that for j #i — 1, we have that Wz is reduced and hence belongs to A* by Lemma 3.2. Also, uv = W001---02100102---dmWm by Lemma 3.4. It follows that (u‘,v‘) € B where u‘ = W001‘-- W0-201-1 and ν' = wTNai ■ ■ ■ amwm. We claim that (u,v) ε (u‘, v‘) N. Indeed, note that (w 1, w 1w 1) € N since ww, 1w = W;-1W 1 = 1, and
(u‘, v)(w 1,0%-1w 1) = (uw 1,0% 1W 1w;-10 • • • amwm) = (u, v),
as required. We conclude that B is a generating set for Μ χ M as a right N-set. It remains, to show eacli element of Μ χ Μ has only one expression in the form (a1,a2)(n1,n2) witli (a1,a2) E B and (n1, 712) € Ν.
Suppose that (αι,α2) € B and let x,y be reduced words with xy = 1. We show that (01,02) can be recovered from the right-hand side of (01,02)(x,y) = (a1x,ya2). Note that by definition of B, 01 has no invertible suffix.
First we claim that @1x is reduced. Indeed, since 01 and x arc reduced, any rewrite rule that is applicable must cut across (01,x). Since the left-hand side of a rewrite rule is nonempty and invertible, this means we can write 01 = οίβ and χ = yb witli Bay invertible and 3, Y nonempty. Since x is right invertible, we deduce that Y is invertible and hence β is invertible, contradicting that oi has 110 invertible suffix. Hence οιτ is reduced.
Consider (a1x,ya2). Let μ be the longest invertible word that cuts across (x1x,yx2) if such a word exists, while if no sucli word exists then let μ be the longest invertible suffix of &1x, which could possibly be empty. We claim that a1xyd2 = @1v witli pv = 02. It will then follow that we can recover (a1, a2) from (a1,a2)(x,y).
Indeed, to prove the claim let 0102 = wo@0 ■ ■ · amWm in Otto-Zhang normal form. Each wj is reduced because 0102 is reduced, by the definition of B, and then wz e A* by Lemma 3.4. Then by definition of B, we have some i with a1 = W000 --- Wt-207-1 and a2 = wi-iai ■ ■ ■ amwm. Then a1xya2 = W000---W1-20i-1xyWi-101---amWm- By Lemma 3.1, it follows that any maximal invertible subword of 01xya2 is either a maximal invertible subword of 01, no or contains xy. But since xy = 1 in M and Wi-1 is maximal invertible in 0102, it follows that xyw;-1 is the maximal invertible subword of aixya2 = W000-·: Wi 20 1xyWi-102---amWm containing xy. It follows that the Otto Zhang normal form of the word d1xya2 is
0117/02 = Woao • · · W;-2a;-1(xyw;-1)a; - - ■ amwm
where the maximal invertible subwords in this normal form are the words
wo,...,Wi-2,xywi-i,wi,...,wm.
Since xywi-i is invertible it follows that ywi-i is left invertible. Since αμυι ■ ■ ■ amwm has no invertible prefix it follows that 110 invertible word cuts across the pair
(ywi-i, αμ^ - - - amwm)-
Indeed, if o = 7102 were an invertible word that cuts across this pair witli 72 a prefix of aiwi ■ ■ ■ amwm, then 01 is left invertible since it is a suffix of the left invertible word ywi-i, and 01 is right invertible since it is a prefix of the invertible word σ. Then σι and σ are invertible and thus 72 is an invertible prefix of aiwi ■ ■ ■ amwm which is a contradiction. It follows that every invertible subword of the word
W;-10;W; * * • ^m^m
is contained in either a subword of ywi-i or else is a subword of aiWi---amwm. It follows that the Otto-Zhang normal form of ya2 is graphically equal to the product of the Otto- Zhang normal form of ywi-i and v := aiw, - ■ ■ amwm. We conclude from Lemma 3.4 that ya2 = ywi-ia^wi ■ ■ ■ amwm. Then
01x702 = wodo - - - W;-2@;-1xyW;-10;W; • • • amwm-
Now there are two cases. If there is an invertible word that cuts across (a2x, yao) then it follows that μ = xywj-i since xy = 1, xywi-i is invertible, and wi-i is a maximal invertible subword of 0102. Therefore, a1yx2 = @1/7 and pv = 02. In the other ease there is no invertible word that cuts across (a1x,ya2). In this case since xywi-1 is invertible, either x is empty or ywi-i is empty. If χ is empty then y is also empty since xy = 1 and χ and y are reduced. Therefore, (a1x, yx2) = (a1, a2), and by definition of B the word 01 has no non-trivial invertible suffix, so μ is empty. So we recover & 2 = 1 = pv. Otherwise, yw —1 is the empty word so
01xy02 = woao · · · W;-2a-1x0;W; · · • dmwm.
In this case in Μ we have χ = xywi-i = Wi-1. In particular x is invertible. Furthermore
@1x = ?T70CZ0 • ■ ■ W1-201-1x
Suppose that χ is not the longest invertible suffix of the word &1x; say, e.g., z is a longer such suffix. Then since xywi-i and z are both invertible with χ a suffix of z, it would follow from Lemma 3.1 that zyw;-1 is an invertible word strictly containing xyWi-i contradicting the fact proved above that xyw;-1 is a maximal invertible subword of a1xya2. Hence in this case χ is the longest invertible suffix of the word @1x. So by definition μ =χ and
Q1xya2 = aixaiwi · · · amwm == @1/v
and
p = xazw; ·· · dmwm = Wi—10W; ■ · ■ amwm = Q2
since in this case, as wc saw above, x = Wi-1 in M. This completes the proof that we can recover (01, α2) from (01, a2)(x,y).
Next suppose that (01, a2) E B and (01, a2)(x, y) = (01, a2)(x‘, y‘) with (x, y) and (x‘, y‘) € N. Wc show that (x, y) = (x‘,y‘). Indeed, by Corollary [24, Corollary 3.8], M is a free right R1-set and a free left L1-set. Note that R1, L1 arc cancellative monoids. Hence the equations aix = a1x‘ and y a2 = y‘a2 imply that x = x‘ and y = y‘, as required.	□
Remark 5.2. Let u,v e A* be reduced. Write
uv = woaiwi...amwm
as in the statement of Lemma 3.3, where a; e A for all 1< i< m and w; is a maximal invertible
subword of w for all 0<i<m (where some of the wi could be the empty word). Suppose that
u = woai ■ • • W;-20;-10f 1
V = Wi-1 azwi ’ ■ ■ &mWm
witli Wi-1 = w1w7-1 The proof of Theorem 5.1 shows that (u,v) is in the weak N-orbit of the basis element (woai · ■ ■ w; 2"; }■ w: μι, ■ ■ ■ amwm).
Since the set Μ χ M is a free right N-set Theorem 5.1, the observation in the previous paragraph provides a characterisation of exactly when two pairs (u1,v1) and (u2,v2) belong to the same weak N-orbit of the free right N-set Μ χ M. Indeed, (u1,v1) and (u2,v2) belong to the same weak N-orbit if and only if the basis elements from B obtained from (u1,v1) and from (u2,v2) via the process described in the previous paragrapli are the same. This description of the weak N-orbits of M χ M will be used in several arguments below.
Recall from the preliminaries section above that if a monoid Ν acts on the right of a set A, then the weak orbits of Ν on A are the equivalence classes under the smallest equivalence relation 011 A sucli that a, an are equivalent for all a e A and η e N. Wc recall for the benefit of the reader that ZA AΛμ ^ is a free abelian group witli basis the set of weak orbits of Ν 011 A via the map sending a @1 to the weak orbit of a.
An important consequence of Theorem 5.1 and Remark 5.2, which will be proved in Lemma 5.3 below, is that Γ /N is a forest, where Γ is the two-sided Cayley grapli of M and Γ /N is the graph obtained by collapsing the induced subgraph on each weak orbit of TV on the vertices of Γ to a point. So Γ /TV has vertex set (M x M)/N and edge set consisting of those edges e of Γ with initial vertex i(e) in a different weak TV-orbit than terminal vertex r(e). The corresponding edge of Γ /TV goes from [i(e)] to [7(e)] where [(m, m‘)] denotes the weak TV-orbit of (m, m‘) e M x M. This gives a bijection between the edges of Γ /TV and tire set of edges of Γ whose initial and terminal vertices belong to distinct N-orbits. Thus for each sucli edge in Γ we can sensibly talk about the corresponding edge in Γ /TV. There is an Μ X M°P-equivariant cellular map T —> T /TV sending a vertex to its class, and sending any edge connecting two vertices in the same weak TV-orbit to that weak TV-orbit (which is a vertex of the graph Γ /TV) and sending every other edge in Γ to the corresponding edge in Γ /TV.
Recall that T0(Γ) denotes the set of path-connected components of Γ whicli is isomorphic to M as an M x M°p-set; see [23, Section 5.3]. Note that if [(m1,m2)] = [(m{,m2)], then m1m2 = m‘{m5. It follows that the map T —> T /TV factors through the map Γ —> T0(Γ ) = M, and so TO (Γ /TV) is identified witli M.
Lemma 5.3. The quotient Γ /N of the two-sided Cayley graph Γ of M by Ν is a forest.
Proof. Let s e Μ and consider the component corresponding to s in Γ /TV. Let w be tire reduced word that equals s. Write
W = W0aWi .. . @mWm
in Otto-Zhang normal form where the w arc maximal invertible subwords (possibly empty, and all in A*) and all a; € A, as in the statement of Lemma 3.3. Let w‘ = 0102...am. We claim that the m-component of Γ /TV is isomorphic to the labelled graph denoted A(w’) with vertex set 0,...,m and an edge labelled by a; from i — 1 to i for 1<i< m. It will then follow that Γ /TV is a forest.
Define a cellular map from the m-component of Γ to A(w’) as follows. If Q1,Q2 € A* are reduced with 0102 = m, then @1 = W0a1 --: w-207 1 1 and Q2 = wh 10;w;--- amwm witli w1w" 1 = Wi-1 by Lemmas 3.3 and 3.4. Both w1 and w 1 can be empty. Send (a1,a2) to i — 1. Notice in particular that the basis clement (woai ■ ■ ■ wi-2ai-i, Wi-iTWi - - ■ amwm) is sent to i — 1, and so this map is surjective on vertices. By Remark 5.2, this map factors through (M X M)/N, and the induced map is injective on vertices. An edge in the m-component of Γ has the form (u,a,) where p, e A* arc reduced, a e A and pay = m. By Lemmas 3.3 and 3.4, we have two cases. The first is that μ = W^Q ■ - rnty^ai-yw'^, μ = w'f_rai■■■amwm where w/i_1aw'._1 = Wi-i. In this case, by Remark 5.2, this edge is between two vertices in the same weak TV-orbit, namely that of the basis element (woao ■ ■ ■ Wi-zai-^wi-iai ■ ■ ■ amwm). The second case is that μ = u>oao ■ ■ ■ wi-i, Y = wi---amwo and a = a2. In this case, initial vertex is in the weak TV-orbit of the basis element (woao ■ ■ ■ ai-i,Wi-iai ■ ■ ■ amwmf whereas the terminal vertex is in the weak TV-orbit of the basis element (W0---W1102,W2---0mWm). We map the corresponding edge of Γ /TV to the edge labeled a; from i — 1 to i. This completes the proof of the isomorphism.	□
6.		Proving THE SET OF COLLAPSED EDGES IS A FINITELY GENERATED FREE M X M°P-SET
As in the previous section, throughout this section
M = (A| w1 = 1,...,w = 1>.
Up to this point we have proved that:
•		N = G * C* for some finite set C where G is the group of units of M.
•		M x Μ is a free TV-set, with basis B defined above, under the action
(m1,m2) - (n1,n2) = (min1,n2m2).
The final thing we need to prove is that the set E of edges collapsed when forming Γ /N is a finitely generated free M x M°p-set. More precisely, wc want to consider the edges E of the two-sided Cayley graph Γ between vertices in the same weak TV-orbit of Μ X M. Then M X M°P acts on this set E of edges via
(mi,m2) · (u, a,v) = (miu, a, vm2)
where (mi,m2) E M X M°P, a E A and u, υ,mi,m2 are reduced words over A.
Drawing an analogy with our argument in [24] the edges E arc the analogue of edges between vertices in the same 2-class in that argument, and TV is playing the role of R1 - the right units. Note however that N is not in general the submonoid of right units of the monoid Μ X M°P.
With the above definitions we want to prove the following.
Theorem 6.1. Let E be the set of edges of the two-sided Cayley graph Γ between vertices in the same weak N-orbit. Then E is a finitely generated free left M χ M°P-set wider the action
(mi,m2) ■ (u,a,v) = (miu, a, vm2)
for (mi,m2) E M x M°P and (u, a,v) E E. with finite basis
C = {(d1,a,02) : d1a82 is invertible, a E A, 81, 02 € A* are reduced, and no proper invertible subword of d1a02 contains the particular occurrence of a that lies between 81 and 82}.
Furthermore, if (81,a,62) E C then διαδ2 E Δ.
Proof. Let (u,a,v) be an edge with u,v E A* reduced, and let
uav = w0a1 · · -amwm
in Otto-Zhang normal form. First of all we claim that (u, a, v) e E if and only if the distinguished occurrence of a is not any of the az. Suppose first that a; is the distinguished occurrence of a for some i, then uav is reduced by Lemma 3.4 (since u, v are reduced, and hence their subwords Wj are reduced) and
(u,av) = (woai ■ ■ ■ aj-iwi-i,ai ■ ■ ■ amwm)	and
(ua, v) = (wo - - ■ Wi_iai,Wiai+1 ■ ■ · amWm)
are in different weak N-orbits by Remark 5.2, as (ua, υ) is a basis element and
(woai ■ ■ ■ ai-i,wi-iai ■ ■ ■amwm)
is the basis element in the weak orbit of (u,αυ).
On the other hand, suppose that a is a letter of some ug. Then u = woai---aw, and υ = w"ai+i ■ ■ • amWm witli w2 = w'aw''. It follows from Remark 5.2 that (ua, v) and (u, av) both belong to the weak TV-orbit of the basis element (W0a1---a2,wai+1---amwm). In particular, eacli element of C belongs to E.
We must now show that if (u,a,v) E E, then (u,a,v) = (m1,m2)(81,a,02) for a unique (mi,m2) E M X M°P and (81,a,82) € C. As before let uav = woai---amwm in Otto-Zhang normal form. Then, by the above, there is a unique i sucli that the distinguished a occurs in ug. Since w2 is invertible, there is a unique minimal invertible subword διαδ2 containing this a by Proposition 4.1. Write u = α'δι and ν = δ^ν'. Then (δι,α,δ2) E C and (u,a,v) = (u',ν')(δι,α,δ2). Wc claim that these choices arc unique.
Clearly the letter a is uniquely determined by (u,a,v). Suppose that a, β, δ(,δ2 arc reduced with (a, B)(8{, a, δ'2) = (u, a,v) and (81, a, 84) E C. Then (u, a, v) = (adl, a, 853). We claim that αδ( and δ!2β are reduced. We handle just the first case as the second is dual. Since a, δ( are reduced and each left-hand side of a rewrite rule is nonempty and invertible, it follows that the only way wc can apply a rewrite rule is if it overlaps a and 81. This would imply that 81 has a left invertible prefix. Now any prefix of 81 is right invertible since dad, is invertible. But then wc obtain a contradiction to the fact that 81 has no invertible prefix by Proposition 4.1. Thus αδ( = Νδι and, dually, 853 = o2v’. Then dlao and διαδ2 arc both minimal invertible subwords of uav = αδ^αδ^β containing the distinguished a. By uniqueness of the minimal invertible subword containing a, which follows from Proposition 4.1, wc conclude that dad, = d1a82, with the distinguished occurrences of a the same. Thus 81 = 81, 02 = 85, u‘ = Q and ν' = β. This establishes the uniqueness, and so C is a basis.
To prove that C is finite we show that if (d1,a, ^) € C, then 01a02 e Δ. We begin by showing that 81a82 E Δ*. If d1a02 is reduced, then it belongs to Δ* by Lemma 3.2 since it is invertible. Otherwise, we can apply a rewrite rule to d1a02. Since 01,02 are reduced, the left-hand side of the rewrite rule must contain a. Since each left-hand side of a rewrite rule is in Δ+, hence invertible, and ^α^ has no proper invertible subword containing the distinguished a, we must have that 81a82 is the left-hand side of a rewrite rule, and hence in A*. It follows now that there is a subword of 81a82 containing a that belongs to Δ. But then since elements of Δ are invertible, we deduce by minimality that 81a82 € Δ. We conclude that C is finite as Δ is finite.	□
7.		TWO-SIDED HOMOLOGICAL FINITENESS PROPERTIES OF SPECIAL MONOIDS
Witli all the results in place they can be combined to prove the following main result. We shall use freely that ZM-bimodules are the same thing as left Z[M x M°P]-modules.
Before giving the proof, we make a general topological remark. If X is a CW complex, then the cellular chain complex can be augmented to
—C1(X) - C(X) - H0(X) - 0
where the map C0(X) —> H0(X) is the projection. This chain complex is natural witli respect to cellular maps. Identifying H0(X) witli the free abelian group on the path components of X, this map sends each vertex (viewed as a basis element of Co(X)) to its patli component. Hence we can view this complex as the direct sum of the standard augmented cellular chain complexes of the path components of X. In particular, it will be exact if each path component of X is contractible.
Theorem 7.1 (Theorem A). Let M be a finitely presented monoid of the form
M = (A | w1 = 1,...,w = 1>
and let G be the group of units of M.
(i)		If G is of type FPn (for 1 < n < 00) then Μ is of type bi-FPn.
(ii)		Furthermore, the Hochschild cohomological dimension of Μ is bounded below by cd(G) and bounded above by max{2, cd(G)}.
Proof. Let Γ be the two-sided Cayley grapli of M and let F be Γ /N whicli is a forest by Lemma 5.3. Notice that the multiplication map Μ χ Μ —> M factors through (Μ χ M)/N and hence the connected components of F arc still isomorphic to Μ as an M-bisct. The augmented cellular chain complex
0 -, C1(F) , C0(F) , H0(F) = ZM , 0
is a bimodule resolution of ΖΜ with exactness following from the discussion before the theorem, as F is a forest and hence has contractible patli components.
Now C1(F) is the quotient of a free bimodule on A, namely Z[M x A x Μ], by ZE where E is the finitely generated free M x M°p-set from Theorem 6.1. Thus wc have a free resolution
0 —>ZE —, Z[M x A x Μ] -, C1(F) -, 0
by finitely generated free bimodules, and so Ci(F) is FP0o and has projective dimension at most 1.
Wc have C0(F) = Z[(M x M)/N] = Z[M χ M] ®ZN Z. In Theorem 4.8 wc proved that N is a free product of the group of units G and a finitely generated free monoid. Together with [24, Corollary 4.13] this then implies that N is of type left-FPn, and hence Z is a left ZN- module of type FP„. Let F, —> Z —> 0 be a free resolution of Z by ZN-modules with F; finitely generated for 0<i< n. Then since M X M°P is a free right N-set, it follows that Z[M X M°P] is free as a right ZN-module and hence flat. Thus Z[M χ M°P] ØZN F. —, Z[M x M°P] ®ZN Z->0
is a free resolution with Z[M x M°P] SZN F; finitely generated for 0 < i < n. Therefore, C0(F) = Z[M χ M°P] OZN Z is a Z[M x M°p]-module of type FPx. Now since C0(F) is FPx and C1(F) is FPæ it follows from Lemma 2.1 that the ZM is FPx as a Z[M x M°p]-module, and hence the monoid M is of type bi-FP,.
By [24, Corollary 4.15], since N is a free product of G and a free monoid we can choose a free resolution of Z over ZN to have length at most max{1,cd(G)}, and so we can deduce from the above argument that C0(F) has projective dimension at most max{1, cd(G)}. Since C1(F) has projective dimension at most 1, we deduce that hM has projective dimension at most max{2, cd(G)}, and thus M has Hochschild dimension at most max{2, cd(G)}, by Lemma 2.2.
The fact that the Hochschild cohomological dimension of M is bounded below by that of G follows from [24, Theorem 3.16(2)] together with the fact that the Hochschild cohomological dimension bounds both the left and right cohomological dimension of a monoid; see [23, Section 7]. □
7.1.		Applying the general result to special one-relator monoids. In the one-relator case the main theorem above specialises to the following which gives a Lyndon’s Identity type theorem for the two-sided homology of one-relator monoids.
Theorem 7.2 (Theorem B). Every one-relator monoid of the form (A\r = 1) is of type bi-FPo. Moreover, if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most 2, while if r is a proper power then the monoid has infinite Hochschild cohomological dimension.
Proof. Let M = (A |r = 1>. The group of units G of M is a one-relator group by Adjan’s theorem [1, Lemma 96] and hence the group G is of type FPx by Lyndon’s Identity Theorem [44]. Now applying Theorem 7.1(i) we deduce that M is of type bi-FP.
For the second statement, if r is not a proper power then the group of units G will be a torsion-free one-relator group (see e.g. [24, Lemma 3.18]). Then Lyndon’s results [44] imply that G has cohomological dimension at most 2, which implies the same for the Hochschild cohomological dimension of Μ by Theorem 7.1(ii).
If r is a proper power then G has torsion and hence infinite cohomological dimension. Therefore M has infinite Hochschild cohomological dimension by Theorem 7.1(ii). □
8.		Other one-relator monoids
In [22] we proved that all one-relator monoids (A |u = v> are of type left-FPæ (and dually of type right-FP). In that paper we also classified the one-relator monoids of cohomological dimension at most 2. It is natural to ask to what extent the results in this paper on two-sided homology might be extended to general one-relator monoids (A | u = v>. In particular, it would be interesting to see whether the techniques in this paper can be further developed to prove that all one-relator monoids (A |u = v> are of type bi-FPo.
One natural approach to answering this question would be to try to use the same general proof strategy that was employed in [22] in the one-sided case. From that viewpoint, the result in this paper showing that special monoids arc all of type bi-FP would provide one important infinite family of base cases. The other infinite family of base cases are the, so-called, strictly aspherical one-relator monoids. It turns out (as wc shall explain below) that known results can be applied to deduce that these strictly aspherical one-relator monoids are also of type bi-FP. Hence combined with the results in this paper that deals with all the base cases. Thus following same approach as in [22] this means that the key to proving that all one-relator monoids arc of type bi-FP will be to develop an understanding of how the property bi-FP behaves under a process called compression (a common framework for both Adjan-Oganesyan compression [2] and Lallement compression [41] that was developed in [22]). Developing this understanding in the one-sided case was already quite an involved task, occupying much of the paper [22], and wc expect the two-sided casc to be even more challenging.
Let us outline this approach in more detail. For full details of the terminology used here wc refer the reader to [22]. Let M be the one-relator monoid defined by the presentation (A | u = v> where, without loss of generality, wc assume v < u. This presentation for M is called compressible if there is a nonempty word r e At such that u,ve A*r η rA*. Otherwise, the presentation is called incompressible. If u,v e A*r η r A* then there is an associated compressed one-relator monoid Mr that lias a sliorter defining relation, and tlie word problem of Mr is equivalent to that of M. The compression operation is transitive and confluent and it follows that there is a unique incompressible monoid M' to which M compresses.
It follows from [22, Lemma 3.4 and Proposition 3.5] and [33, Corollary 5.6] that if M is the one-relator monoid defined by the presentation <A | u = v> and M' is the unique incompressible monoid to which M compresses then either
(i)		M’ is a special one-relator monoid <B |w = 1>, or else
(ii)		M’ is a non-special one-relator monoid with a strictly aspherical presentation <B|w = z).
Here a monoid presentation is strictly aspherical if each connected component of the Squier complex of the presentation is simply connected (see [22, Section 2] for the relevant definitions).
In case (i) above it follows from the results in this paper that M‘ = (B | w = 1> is of type bi-FPæ. In case (ii) the compressed monoid M' = <B | w = z> is strictly aspherical, and it turns out, as we shall now explain, that this is enough to show that M' is also of type bi-FPo. Indeed, a key fact used in the proof of the main result of [22] was that the Cayley complex of a strictly aspherical presentation turns out to be contractible; see [22, Lemmas 6.4 and 6.5]. It turns out that key fact also holds in the two-sided case too, and that it is likely to be equally important in the approach we are describing here to proving that all one-relator monoids are of type bi-FPoo.
In more detail, in [54] Pride defines π^\ρ) = H1(D(P)) where D(P) is the Squier complex of the presentation. Hence if Ρ is strictly aspherical then ni(D(P)) is trivial at every vertex, hence in this case 7b)(P) = H1(D(P)) = 0. Then the argument in the final two paragraphs of the article [57] shows that the second homology of the 2-sided Cayley 2-complex of the presentation P is equal to 7§b)(P). When the presentation P is strictly aspherical this implies that second homology of the two-sided Cayley complex is 0. Together with the fact proved in [23, Theorem 7.12] that each connected component of the two-sided Cayley complex is simply connected, this implies by the Hurewicz and Whitehead theorems that the connected components of the two-sided Cayley complex are contractible. Then it follows that the two-sided Cayley graph is a bi-equivariant classifying space for the monoid, in the sense defined in [23, Section 7]. from which it follows that the monoid is bi-Fx and hence by [23, Proposition 7.7] is also bi-FP2. Also this shows that a strictly aspherical finitely presented monoid has Hochschild cohomological dimension at most 2, again by results of [23, Section 7], since the argument above shows that the 2-sided Cayley 2-complex is a bi-cquivariant classifying space for the monoid.
In particular the observations in the previous paragraph can be used to give an alternative proof that when w is not a proper power the torsion-free special one-relator monoid (A w = 1> is of type bi-FP and has Hochschild cohomological dimension at most 2, since Kobayashi proved in [34, Corollary 7.5] that torsion-free special one-relator monoids are strictly aspherical. In addition these observations show the property bi-FPæ holds for one-relator monoids also in the following cases.
Theorem 8.1. Let M = <A |u = v> be an incompressible one-relator monoid with u,v e At, that is, there is no nonempty word r e At such that u,v e A*r arA*. Then Μ is of type bi-FP00 and the Hochschild cohomological dimension at most 2.
The class of examples covered by the previous proposition should, from the discussion in this section above, be viewed as a family of base cases for a possible future inductive proof using compression that all one-relator monoids arc of type bi-FPo.
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