Showing papers on "Bicyclic semigroup published in 1990"
65 citations
TL;DR: In this paper, it was shown that the type-II-construct subsemigroup of a finite semigroup S is regular, and that if the idempotents in S commute, then S divides a finite inverse semigroup.
Abstract: We prove that if the "type-II-construct" subsemigroup of a finite semigroup S is regular, then the "type-II" subsemigroup of 5 is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results: (1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute. (2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup. (3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety UQ * G if and only if Sn belongs to UQ. Here UQ denotes the pseudo-variety of finite semigroups which are unions of groups. For these three classes of semigroups, type-// is equal to type-// construct.
27 citations
TL;DR: In this article, the structure of inverse semigroups and associated C* algebras were discussed, and it was shown that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).
Abstract: We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).
24 citations
18 citations
12 citations
9 citations
7 citations
TL;DR: In this paper, the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semiigroups is investigated, and a convenient description of the greatest idempotent-separating congruence on (S ) is given.
Abstract: For an inverse semigroup S , the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by ( S ) and called the partial automorphism monoid of S Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that ( S ) is completely semisimple Secondly, for an inverse semigroup S , we find a convenient description of the greatest idempotent-separating congruence on ( S ), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental
6 citations
01 Jan 1990
TL;DR: In this paper, it was shown that a regular semigroup S C 9 (F) has the permutation property 37, m > 2, if for every a,..., am E S, there exists a non-trivial permutation a such that al a = a,()... aU(,nf).
Abstract: It is well-known that if a semigroup algebra K[S] over a field K satisfies a polynomial identity then the semigroup S has the permutation property. The converse is not true in general even when S is a group. In this paper we consider linear semigroups S C 9 (F) having the permutation property. We show then that K[S] has a polynomial identity of degree bounded by a fixed function of n and the number of irreducible components of the Zariski closure of S. A semigroup S is said to have the property 37, m > 2, if for every a,, .. ., am E S, there exists a non-trivial permutation a such that al a = a,()... aU(,nf). S has the permutation property 37 if S satisfies 3Y for some m> 2. The class of groups of this type was shown in [3] to consist exactly of the finite-by-abelian-by-finite groups. For the recent results and references on this extensively studied class of groups, we refer to [1]. The above description of groups satisfying 37 was extended to cancellative semigroups in [11], while a study of regular semigroups with this property was begun in [6]. In connection with the corresponding semigroup algebras K[S] over a field K, the problem of the relation between the property 37 for S and the Plproperty for K[S] attracted the attention of several authors. It is straightforward that S has 37 whenever K[S] satisfies a polynomial identity. However the converse fails even for groups in view of [3] and the characterization of PI group algebras, cf. [1 5]. On the other hand, K[S] was shown to be a PI-algebra whenever S is a finitely generated semigroup (satisfying 3 ) of one of the following types: periodic [20], cancellative [11], 0-simple [3, 5], inverse, or a Rees factor semigroup of free semigroup, cf. [12]. However, a finitely generated regular semigroup S with two non-zero OF-classes having Y but with K[S] not being PI was constructed in [12]. The main result of this paper is that if S is a linear semigroup satisfying 39, then K[S] is PI for any field K. In the course of the proof, we obtain a structural description of a strongly 7r-regular semigroup of this type. The basic technique is to consider the Zariski closure S of S. Then S is a linear Received by the editors July 7, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20M25, 16A38; Secondary 20M20, 16A45. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page
6 citations
01 Nov 1990
TL;DR: In this article, it was shown that for the Baer and Levitsky radicals of commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].
Abstract: Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S , the equalities B ( RS ) = B ( R ) S and L ( RS ) = L ( R ) S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π( RS ) = π( R ) S , where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].
5 citations
TL;DR: In this article, the authors define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.
Abstract: In this paper we define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.
TL;DR: A ring (R,*) with involution * is called formally complex if implies that all Ai are 0 as discussed by the authors, and a semigroup ring (S, *) with proper involution is a formally complex ring.
Abstract: A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.
01 Jan 1990
TL;DR: A semigroup in C will frequently be called a C-semigroup as discussed by the authors, which is a class of semigroups with the property that T ≈ S implies T ∈ C.
Abstract: By an abstract class of semigroups we shall mean a class C with the property that S ∈ C and T ≈ S implies T ∈ C. A semigroup in C will frequently be called a C-semigroup.
01 Jan 1990
TL;DR: The subject of abstract ideal theory has developed along two more or less distinct though related lines since its beginnings in the twenties and thirties as mentioned in this paper, one line involves an ideal operator x which is applied to subsets of a semigroup S to produce a lattice ordered semigroup L of ideals.
Abstract: The subject of abstract ideal theory has developed along two more or less distinct though related lines since its beginnings in the twenties and thirties. One line involves an ideal operator x which is applied to subsets of a semigroup S to produce a lattice ordered semigroup L of ideals. The other, and the one of primary interest to us in this discussion, begins immediately with the lattice ordered semigroup L, dispensing entirely with the assumption of underlying elements.
TL;DR: In this paper, one-sided identity elements and subsets of the semigroup of all binary relations on a nonempty set are described and formulas for the number of identities for a finite set are given.
Abstract: We describe one-sided identity elements and subsets of the semigroup of all binary relations on a nonempty set. We obtain formulas for the number of identities for a finite set.
01 Jan 1990
TL;DR: In this paper, it was shown that a *-semigroup *-generated by a family of commuting Hilbert space isometries that commute each other, none of which commutes with the adjoint of another one, and none of whose power is a nonzero power of another power, is a singly *-generative semigroup.
Abstract: It is shown that if a *-semigroup *-generated by a family of commuting Hilbert space isometries that commute each other, none of which commutes with the adjoint of another one, and none of which is a nonzero power of another one, consists of partial isometries, then it is singly *-generated. Also, the following result on algebraic semigroups is proved: If S is an inverse semigroup *-generated by a set X satisfying the generating relations: a* a = 1 ab = ba, for all a, b e X, then S is the bicyclic semigroup. Both results follow from the special behavior of inverse *-semigroups *-generated by analytic Toeplitz operators.
TL;DR: In this paper, a simple proof is given that the infinitesimal generator of the heat semigroup and the Poisson semigroup are scalar operators in Lp (Rn), 1
TL;DR: In this article, the authors studied combinatorial properties of an infinite word on a two-letter alphabet in order to provide a solution to a problem of Semigroup Theory, which they called Semigroup Problem.
Abstract: We study some combinatorial properties of an infinite word on a two letters alphabet in order to provide a solution to a problem of Semigroup Theory.