Locally adequate semigroup algebras

Ji, Yingdan; Luo, Yanfeng

doi:10.1515/math-2016-0004

Cited by 10 publications

(3 citation statements)

References 26 publications

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“…In the following, we provide two corollaries of Theorem 3.8. is strongly nil-clean for each ∈ ( / ) * α S . Furthermore, for each ∈ ( / ) * α S , the maximal subgroup of D α 0 is isomorphic to the maximal subgroup in D α , and the number of -classes (resp., -classes) in D α 0 is equal to the number of -classes (resp., -classes) in D α , see [13]. The result then follows from Theorem 3.8.…”

Section: Lemma 32 [2]mentioning

confidence: 74%

The strong nil-cleanness of semigroup rings

2020

Open Mathematics

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In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

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Section: Lemma 32 [2]mentioning

confidence: 74%

The strong nil-cleanness of semigroup rings

2020

Open Mathematics

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“…Lawson proved [12] that the category of all E-Ehresmann semigroups is isomorphic to the category of all Ehresmann categories. In this paper we discuss algebras of these objects over some commutative unital ring K. In recent years, algebras of semigroups related to Ehresmann semigroups have been studied by a number of authors, see [8,9,11].…”

Section: Introductionmentioning

confidence: 99%

Algebras of Ehresmann semigroups and categories

Stein

2016

Semigroup Forum

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E-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an EEhresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize E-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.

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“…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%

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

Margolis¹,

Stein²

2020

Preprint

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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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Locally adequate semigroup algebras

Cited by 10 publications

References 26 publications

The strong nil-cleanness of semigroup rings

The strong nil-cleanness of semigroup rings

Algebras of Ehresmann semigroups and categories

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

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