Two-sided homological properties of special and one-relator monoids

Gray, Robert D. and Steinberg, Benjamin (2026) Two-sided homological properties of special and one-relator monoids. Forum of Mathematics, Sigma. ISSN 2050-5094 (In Press)

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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 dened 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-FP8. 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 with defining relation u = v we show that if there is no nonempty word w such that u, v e A* w n wA* then M is of type bi-FPx and the Hochschild cohomological dimension at most 2.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups:	Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR)
Depositing User:	LivePure Connector
Date Deposited:	18 Mar 2026 10:30
Last Modified:	18 Mar 2026 10:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/102394
DOI:	issn:2050-5094

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