Twisted products of monoids

East, James, Gray, Robert, Muhammed, P. A. Azeef and Ruškuc, Nik (2025) Twisted products of monoids. Journal of Algebra. ISSN 0021-8693

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Abstract

A twisting of a monoid S is a map Φ : S × S → N satisfying the identity Φ(a, b) + Φ(ab, c) = Φ(a, bc) + Φ(b, c). Together with an additive commutative monoid M, and a fixed q ∈ M, this gives rise a so-called twisted product M × q Φ S, which has underlying set M × S andmultiplication (i, a)(j, b) = (i + j + Φ(a, b)q, ab). This c onstruction has appeared in the special cases where M is N or Z under addition, S is a diagram monoid (e.g. partition, Brauer or Temperley-Lieb), and Φ counts components in concatenated diagrams. In this paper we identify a special kind of ‘tight’ twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green’s relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Schützenberger groups, and more. We also a number of examples, including severalapparently new ones, which take as their starting point certain generalisations of Sylvester’s rank inequality from linear algebra.

Item Type:	Article
Uncontrolled Keywords:	twistings,twisted products,diagram monoids,linear monoids,independence algebras
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:	29 Oct 2025 13:30
Last Modified:	30 Oct 2025 01:08
URI:	https://ueaeprints.uea.ac.uk/id/eprint/100833
DOI:	10.1016/j.jalgebra.2025.10.030

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