Twisted products of monoids

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 and multiplication (i, a)(j, b) = (i + j + Φ(a, b)q, ab). This construction 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 several apparently new ones, which take as their starting point certain generalisations of Sylvester’s rank inequality from linear algebra.