On algebras of P-Ehresmann semigroups and their associate partial semigroups

Wang, Shoufeng

doi:10.1007/s00233-017-9903-4

Cited by 15 publications

(6 citation statements)

References 19 publications

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“…This theorem is a generalization of several results due to Solomon [26], Steinberg [33], Guo and Chen [11] and the second author [29]. It was further generalized by Wang [37] to the class of P -Ehresmann and right\left P -restriction semigroups. P -Ehresmann and P -restriction semigroups were introduced by Jones in [16].…”

Section: Projective Modules Of Right Restriction Ei-ehresmann Semigroupsmentioning

confidence: 58%

“…However, Shoufeng Wang has found out that the proof implicitly assumes it. In fact, the functions ψ and ϕ of Theorem 6.4 are k-algebra homomorphisms if and only if S is right restriction [37,Lemma 4.3]. This led to a correction [31], but it was not clear whether being right restriction is necessary for the conclusion of Theorem 6.4 to be true.…”

Section: Appendix: Ehresmann Semigroups and Categories -Two Counterex...mentioning

confidence: 99%

“…A very successful way to study algebras of inverse semigroups or their generalizations is to relate them to algebras of some associated category (or a "partial semigroup"). This is often done with some appropriate Möbius function as in [10,11,15,26,29,33,34,37]. In particular, the second author has proved in [30,31] the following theorem.…”

Section: Introductionmentioning

confidence: 99%

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Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

Margolis¹,

Stein²

2020

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We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose HE(e)class is a group for every projection e ∈ E. This means that its corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite Ehresmann semigroups whose categories are EI is a pseudovariety and we show in the infinite case, that the collection of Ehresmann semigroups whose categories have endomorphism monoids each having one idempotent is a quasivariety. We prove that the simple modules of the semigroup algebra kS (over any field k) are induced Schützenberger modules of the maximal subgroups of S. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones.As a natural example we consider the monoid PT n of all partial functions on an n-set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.

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Section: Projective Modules Of Right Restriction Ei-ehresmann Semigroupsmentioning

confidence: 58%

Section: Appendix: Ehresmann Semigroups and Categories -Two Counterex...mentioning

confidence: 99%

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Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

Margolis¹,

Stein²

2020

Preprint

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“…This result has led to several applications regarding semigroups of partial functions [18,21,22,13] and recently also to the study of certain partition monoids [3]. We mention also that Wang [26] generalized the above results further to a certain class of right P -restriction, P -Ehresmann semigroups (for definitions of these notions see [10]) -but we do not follow this approach in this paper. A hint for another direction is given by the Catalan monoid.…”

Section: Introductionmentioning

confidence: 99%

Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

Stein¹

2021

Preprint

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Let S be a reduced E-Fountain semigroup. If S satisfies the congruence condition, there is a natural construction of a category C associated with S. Under a few conditions, we prove that there is an isomorphism of algebras kS ≃ kC (where k is any unital commutative ring) if some weak form of the right ample identity holds in S. This gives a unified generalization for a result of the author on right restriction E-Ehresmann semigroups and a result of Margolis and Steinberg on the Catalan monoid.

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“…In [25,Section 5] the author proved that each one of the monoid algebras k PO n , k PF n and k PC n is isomorphic to a category algebra of some corresponding subcategory of E n . These are actually examples of an isomorphism that holds for a larger class of semigroups (for details, see [24,26,31]). In [25,Section 5] this isomorphism was used in order to describe the ordinary quiver of these algebras.…”

Section: Introductionmentioning

confidence: 99%

Representation theory of order-related monoids of partial functions as locally trivial category algebras

Stein¹

2018

Preprint

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In this paper we study the representation theory of three monoids of partial functions on an n-set. The monoid of all order-preserving functions (i.e., functions satisfying f (x) ≤ f (y) if x ≤ y) the monoid of all order-decreasing functions (i.e. functions satisfying f (x) ≤ x) and their intersection (also known as the partial Catalan monoid). We use an isomorphism between the algebras of these monoids and the algebras of some corresponding locally trivial categories. We obtain an explicit description of a quiver presentation for each algebra. Moreover, we describe other invariants such as the Cartan matrix and the Loewy length.

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On algebras of P-Ehresmann semigroups and their associate partial semigroups

Cited by 15 publications

References 19 publications

Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

Representation theory of order-related monoids of partial functions as locally trivial category algebras

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