Presentations for singular wreath products

Abstract

For a monoid M and a subsemigroup S of the full transformation semigroup T n , the wreath product M ≀ S is defined to be the semidirect product M n ⋊ S, with the coordinatewise action of S on M n . 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 T n , 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.

On a class of semigroup products

“…• Theorem 6.50 gives a presentation for a semidirect product U ⋊ S in the case that S is a monoid and U an arbitrary semigroup. This complements [41,Theorem 3.1], which treats the reverse case, where U is a monoid and S a semigroup. By contrast, [74, Corollary 2] gives a presentation for U ⋊ S when it is a monoid (which occurs when the conditions discussed in the previous point hold).…”

“…The case in which U is a monoid, and the semigroup S acts on U (= U 1 ) by monoid morphisms, is considered in [41], where a presentation for U ⋊ S is constructed from a presentation for S and the entire multiplication table of U . As we will see in Section 6.5, similar presentations exist in the case that S is a monoid with a monoidal action on U 1 ; see Theorem 6.50.…”

“…We have already noted in Remark 6.2 that presentations for U ⋊ S exist in the case that S acts on U (= U 1 ) by monoid morphisms [41]: i.e., when the pair (U, S) is strong. However, since the presentation from [41] utilises the entire multiplication table for U , attempting to use this (and Proposition 6.1) to obtain presentations for U S ∼ = (U ⋊ S)/θ in this case is undesirable. It also does not apply to the more general situation in which (U, S) is not strong.…”

“…As discussed above, presentations are extremely important tools when working with any kind of algebraic structure, and several results exist for building presentations for algebras that arise from others via natural constructions. See for example [21,24,41,65,74,[105][106][107]; the introduction to [36] contains a fuller discussion and many more references. Chapter 6 contains many general results on presentations for a semigroup U S arising from an action pair (U, S) in several important cases:…”

Presentations for wreath products involving symmetric inverse monoids and categories

“…The situation for M ≀ Sing(I n ) is rather more complex, as Sing(I n ) is not a monoid, and it is well known that semigroups tend to behave quite badly even for much simpler constructions such as cartesian/direct products [15,32,67,68,80]. As a case in point, an entire 41-page paper [34] has been devoted to the related wreath product M ≀ Sing(T n ), where T n is the full transformation monoid (consisting of all self-maps of n). While the forthcoming paper [12] deals with a vast class of semigroup products, and proves many general results, examples such as M ≀ Sing(I n ) were singled out as a special kind of difficult case, and none of the results proved in [12] apply here.…”