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×_Φ^qS, 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 floating 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 consider 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.