Showing papers on "Bicyclic semigroup published in 1995"
Book•
01 Jan 1995
TL;DR: Inverse semigroups as discussed by the authors are a subclass of regular semigroup classes and can be seen as semigroup amalgamations of semigroup groups, which is a special case of regular semiigroups.
Abstract: 1. Introductory ideas 2. Green's equivalences regular semigroups 3. 0-simple semigroups 4. Completely regular semigroups 5. Inverse semigroups 6. Other classes of regular semigroups 7. Free semigroups 8. Semigroup amalgams References List of symbols
1,979 citations
TL;DR: In this article, the existence of the bifree locally inverse semigroup BF L I (X) on a set X is proved by using canonical forms of the elements of the Bifree Completely Simple Semigroup BF C S (X).
18 citations
TL;DR: The weak closure of the image of the Weil representation of the infinite-dimensional symplectic group over a non-Archimedean field of an odd residue characteristic was described in this paper.
16 citations
TL;DR: In this paper, the existence of trace functions on semigroup algebras was studied and necessary and sufficient conditions for trace functions to exist on the semigroup algebra F[S] of an inverse semigroup 5 over a subring F of C that contains 1 and is closed under complex conjugation.
Abstract: Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and 5 is completely semisimple of a type that includes countable free inverse semigroups. The concept of a trace function on a real or complex algebra had its origin in matrix theory and is of central importance in many algebraic and analytical contexts. In the case of a group algebra, the trace of an element is defined simply to be the coefficient of the identity and is easily seen to possess all the standard properties. With the growth of interest in inverse semigroups (a class of involution semigroups with many group-like features), it is natural to ask whether the corresponding semigroup algebras also admit trace functions. In this paper we consider the semigroup algebra F[S] of an inverse semigroup 5 over a subring F of C that contains 1 and is closed under complex conjugation. In Section 1, where the basic definitions appear, two simple necessary conditions are obtained for the existence of a trace function on F[S] and attention is drawn to those trace functions (called 'strong') with the property that their values on the idempotents of S are positive integers. The main result of Section 2 provides a necessary and sufficient condition for F[S] to admit a strong trace function - namely that each principal ideal of the semilattice of S be finite. Section 3 comprises two examples. The notion of a pseudotrace function relative to a submodule is introduced in Section 4 and it is shown that, for any nonempty finite subset T of 5, F[S] admits a
16 citations
TL;DR: It is shown that every semigroup pseudovarieties containing a group not in the subpseudovariety generated by all idempotent generated members of has no finite basis of pseudoidentities provided the five-element idem Potent generated 0-simple semigroup lies in .
Abstract: We show that every semigroup pseudovariety containing a group not in the subpseudovariety generated by all idempotent generated members of has no finite basis of pseudoidentities provided the five-element idempotent generated 0-simple semigroup lies in . This gives, in particular, a counterexample to a conjecture by J. Almeida.
15 citations
TL;DR: In this paper, it was shown that any regular semigroup is a homomorphic image of a regular semiigroup whose least full self-conjugate subsemigroup is unitary; the homomorphism is injective on the sub-semigroup.
12 citations
TL;DR: In this paper, the factor power of a transformation semigroup (S, M) is assigned to a semigroupFP(S) called the factor-power of the semigroup S, M, and applied to the symmetric group Sn.
Abstract: To a transformation semigroup (S, M) we assign a new semigroupFP(S) called the factor-power of the semigroup (S, M). Then we apply this construction to the symmetric groupSn. Some combinatorial properties of the semigroupFP(Sn) are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of ordern and the structure of its idempotents. The idempotents are used in characterizingFP(Sn) as an extremal subsemigroup of the semigroupBn of all binary relations of ann-element set and also in the proof of the fact thatFP(Sn) contains almost all elements ofBn.
11 citations
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TL;DR: In this article, a countable directed family of semigroup congruences is introduced, and a theory analogous to the theory of normal series for groups is developed, which is an effective tool for studying the structures of the lattices formed by certain species of semigroups.
Abstract: A countable directed family of semigroup congruences is introduced, and a theory analogous to the theory of normal series for groups is developed. This rather simple approach, surprisingly, is an effective tool for studying the structures of the lattices formed by certain species of semigroups (classes of semigroups closed under taking homomorphic images) such as varieties, pseudovarieties, and existence varieties etc.
10 citations
TL;DR: In this article, it was shown that the set of all 0-consistent ideals of an arbitrary semigroup with the zero forms a complete atomic Boolean algebra whose atoms are summands in the greatest orthogonal decomposition of this semigroup.
Abstract: The purpose of this paper is to prove that every semigroup with the zero is an orthogonal sum of orthogonal indecomposable semigroups. We prove that the set of all 0-consistent ideals of an arbitrary semigroup with the zero forms a complete atomic Boolean algebra whose atoms are summands in the greatest orthogonal decomposition of this semigroup.
9 citations
9 citations
TL;DR: In this paper, all isomorphisms between comutative semigroup algebras K[S] and K[T] are found when K is a field and S, T have two generators subject to a single homogeneous defining relation.
Abstract: All isomorphisms between comutative semigroup algebras K[S] and K[T] are found when K is a field and S, T have two generators subject to a single homogeneous defining relation
TL;DR: In this paper, the symmetric group and inverse semigroup onn symbols and a semigroup T⊂Cn is considered to be Sn-normal if α−1Tα ⊂T for every α∈Sn.
Abstract: IfSn andCn denote, respectively, the symmetric group and inverse semigroup onn symbols, thenSn⊂Cn and a semigroupT⊂Cn isSn-normal ifα−1Tα ⊂Tfor every α∈Sn. TheSn-normal semigroups are classified.
Journal Article•
TL;DR: In this paper, a simple proof of a known result is given: every inverse semigroup can be isomorphically embedded in the semigroup of cosets of a group, which is a semigroup that can be defined as a group of semigroups.
Abstract: In his seminal article of 1941, Paul Dubreil introduced \textit{complexes forts} of semigroups. Strong subsets of a semigroup $S$ form another semigroup under a natural multiplication. Properties of this semigroup are studied and some open problems raised (specially when $S$ is a group or an inverse semigroup). Also, a simple proof of a known result is given: every inverse semigroup can be isomorphically embedded in the semigroup of cosets of a group.
Journal Article•
TL;DR: In this paper, the multiplicative semigroup of Z_m is studied and the problem concerning the powers of a subset of the semigroups of Zm(m) is studied.
Abstract: In several papers Paul Dubreil has studied the multiplicative semigroup of several types of rings and conditions for semigroups which can serve as an underlying semigroup of a ring. In this paper we shall deal with the multiplicative semigroup of $Z_m$, which will be denoted by $S(m)$, and we shall treat a rather unconventional problem concerning the powers of a subset of $S(m)$.
TL;DR: A sequence of lemmas leads to a two-fold characterisation of the syntactic monoid in the title as mentioned in this paper, which is a characterisation that can be used to characterize the syntactically monoid.
Abstract: A sequence of lemmas leads to a two-fold characterisation of the syntactic monoid in the title. Some alternatives as well as special cases, in particular when the code consists of a singleton, are considered.
TL;DR: In this paper, it was shown that if S is a semilattice of completely 0-simple semigroups and completely simple semiigroups, then the semigroup ring RS possesses an identity iff so does R(E(S)) for the subsemigroup of S generated by E(S).
Abstract: LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.