Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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01 Apr 1992
TL;DR: In this article, it was proved that the semigroup algebra K[M] is the direct sum of n + 1 algebras, namely, of one full matrix algebra over each of the group algesbras K[GL(r, F)] with r = 0, 1,..., n.
Abstract: Let M denote the multiplicative semigroup of all n-by-n matrices over a finite field F and K a commutative ring with an identity element in which the characteristic of F is a unit. It is proved here that the semigroup algebra K[M] is the direct sum of n + 1 algebras, namely, of one full matrix algebra over each of the group algebras K[GL(r, F)] with r = 0, 1, ..., n . The degree of the relevant matrix algebra over K[GL(r, F)] is the number of r-dimensional subspaces in an n-dimensional vector space over F. For K a field of characteristic different from that of F, this result was announced by Faddeev in 1976. He only published an incomplete sketch of his proof, which relied on details from the representation theory of finite general linear groups. The present proof is self-contained.
21 citations
TL;DR: In this paper, it was shown that semigroup algebras of finite ample semigroups have generalized triangular matrix representations, and the structure of the radicals of the semigroup of such semigroup is determined.
Abstract: An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.
21 citations
TL;DR: In this paper, the authors characterize when a Cayley automata semigroup is finite, is free, is a left zero semigroup, a right zero semigroup, or is a group.
Abstract: In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite.
21 citations
TL;DR: A finitely generated semigroup S given by generators and the set of relations between these generators is called an F-semigroup if it is isomorphic to a subsemigroup of a free semigroup as discussed by the authors.
Abstract: A finitely generated semigroup S given by generators and the set of relations between these generators is called an F-semigroup if it is isomorphic to a subsemigroup of a free semigroup. In this paper three theorems concerning F-semigroups are proven and some unsolved problems are presented.
20 citations
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TL;DR: In this paper, the authors give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication.
Abstract: We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial "folding" operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are J-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.
20 citations