Presentations for wreath products involving symmetric inverse monoids and categories

Clark, Chad; East, James E.

doi:10.48550/arxiv.2204.06992

Cited by 2 publications

(6 citation statements)

References 39 publications

(53 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Again, there is the obvious analogue of the story in type B, explored in Subsections 5.4.4 -5.4.7. This includes a presentation for the signed rook monoid in Proposition 20, see also [EF13] and [CE22] for further generalizations.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Finite quotients of singular Artin monoids and categorification of the desingularization map

Jonsson¹,

Mazorchuk²,

Westin³

et al. 2022

Preprint

View full text Add to dashboard Cite

We study various aspects of the structure and representation theory of singular Artin monoids. This includes a number of generalizations of the desingularization map and explicit presentations for certain finite quotient monoids of diagrammatic nature. The main result is a categorification of the classical desingularization map for singular Artin monoids associated to finite Weyl groups using BGG category O.

show abstract

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Finite quotients of singular Artin monoids and categorification of the desingularization map

Jonsson¹,

Mazorchuk²,

Westin³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…As another relevant comparison, the 41-page paper [41] is essentially devoted to a single semigroup of the form U ⋊ S, with U a monoid and S a non-monoid semigroup (the opposite configuration to that covered by Theorem 6.50). A number of wreath products including M ≀ Sing(I n ) are treated in [15], using entirely different methods. We leave it as an open problem to investigate the above-mentioned applications of Theorem 6.50.…”

Section: Two Submonoids IIImentioning

confidence: 99%

“…Wreath products of this kind (and similar) have been studied by numerous authors. See for example [11,15,18,[69][70][71][72]79,85,109]. The introduction to [70] discusses some of the early history of the idea, going back to the work of Specht [110].…”

Section: Presentationsmentioning

confidence: 99%

“…Presentations for the first two (for finite X) are stated above in Theorems 9.32 and 9.33. For the third (again for finite X), see [15].…”

Section: One Submonoid IImentioning

confidence: 99%

“…The wreath products M ≀ Sing(T n ) and M ≀ Sing(I n ) were treated in [15,41], and M ≀ Sing(PT n ) in Theorem 9.33. For the remaining wreath products listed above, we need presentations for M n , M n 0 , G n and T n .…”

Section: One Submonoid IImentioning

confidence: 99%

See 2 more Smart Citations

On a class of semigroup products

Carson¹,

Dolinka²,

East³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

This paper concerns a class of semigroups that arise as products U S, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U 1 that is suitably compatible with the product in the over-monoid.The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest.The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup U S is a quotient of a semidirect product U ⋊S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups.The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for U S out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

show abstract

Presentations for wreath products involving symmetric inverse monoids and categories

Cited by 2 publications

References 39 publications

Finite quotients of singular Artin monoids and categorification of the desingularization map

Finite quotients of singular Artin monoids and categorification of the desingularization map

On a class of semigroup products

Contact Info

Product

Resources

About