Ranks of ideals in inverse semigroups of difunctional binary relations

The set Dn of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (Publ Math Debrecen 78(2):253–282, 2011), which asks: What is the rank (smallest size of a generating set) of Dn? Specifically, we show that the rank of Dn is B(n)+n, where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of Dn. Although Dn bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of J-classes), we note that the fast growth of rank(Dn) as a function of n is a property not shared with these other families

Presentations for singular wreath products

For a monoid M and a subsemigroup S of the full transformation semigroup Tn, the wreath product M≀S is defined to be the semidirect product Mn ⋊S, with the coordinatewise action of S on Mn. The full wreath product M≀T n is isomorphic to the endomorphism monoid of the free M-act on n generators. Here we are particularly interested in the case that S=Sing n is the singular part of Tn, consisting of all non-invertible transformations. Our main results are presentations for M≀Sing n in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that M≀Sing n is idempotent-generated if and only if the set M/L of L-classes of M forms a chain under the usual ordering of L-classes, and we give a presentation for M≀Sing n in terms of idempotent generators for such a monoid M. Among other results, we also give estimates for the minimal size of a generating set for M≀Sing n, as well as exact values in some cases (including the case that M is finite and M/L is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent-generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set