Showing papers on "Bicyclic semigroup published in 1976"
01 Jan 1976
TL;DR: In this article, it was shown that almost all semigroups on n elements have the following property: the n elements are split into sets A, B and there is an e E Bsothatwheneverx,y E A,xy E B,butifxoryisinB,xy = e.
Abstract: The number of semigroups on n elements is counted asymptotically for large n. It is shown that "almost all" semigroups on n elements have the following property: The n elements are split into sets A, B and there isan e E Bsothatwheneverx,y E A,xy E B,butifxoryisinB,xy = e. 1. The problem. Fix a labelled n-element set [n] = (1, ... , n}. A semigroup SG on [n] is an associative binary operation (denoted by concatenation). Let S(n) denote the number of semigroups on [n]. We find an asymptotic approximation to S(n). Let (1.1) f(t) = ( )t+(n-t)2
48 citations
TL;DR: In this paper, the star representations on Hubert space of the inverse semigroup are studied, and specific examples of star representations are constructed for the /'-algebra of a number of the most important examples of inverse semigroups.
Abstract: In this paper the star representations on Hubert space of the /'-algebra of an inverse semigroup are studied. It is shown that the set of all irreducible star representations form a separating family for the /'-algebra. Then specific examples of star representations are constructed, and some theory of star representations is developed for the /'-algebra of a number of the most important examples of inverse semigroups. Introduction. Let 5 be a semigroup (as defined in [2, p.l]). If a, b E S, we write ab for the semigroup product of a with b. Let ll(S) be the set of all complex-valued functions fon S such that 11/11, = L l/(a)l<~. aGS lif.gE /'(S), then the convolution product / * g is given by the definition (f*gXc)= Z f(fl)s(b), cES. a,b with ab=c With convolution multiplication and norm II • II1, I1 (S) is a Banach algebra. If a E S, we identify a with the function which takes the value 1 at a and is 0 everywhere else. In this way S is embedded in /'(S). Having made this identification, when /G P(S) we have /= £ f(a)a. A map a —*■ a* of S into S is called an involution on S if (ab)* = b*a* all a, b E S, and (a*)* a all a G S. If S has an involution *, then P(S) has an involution * defined by the rule /* = E /(a)*«*, feHs), a£5 Received by the editors September 12, 1974. AMS (MOS) subject classifications (1970). Primary 43A65; Secondary 43A20.
48 citations
47 citations
TL;DR: Theorem 1.3 as mentioned in this paper shows that any proper inverse semigroup P can be embedded in a semidirect product P of a semilattice and a group, where P is bisimple with identity.
46 citations
TL;DR: In this article, the semigroup and adjoint semigroup for a class of linear functional differential equations with infinite delays were considered, which includes certain linear Volterra integro-differential equations.
Abstract: We consider the semigroup and adjoint semigroup for a class of linear functional differential equations with infinite delays, which includes certain linear Volterra integro-differential equations. In particular, we show that by an appropriate choice of the state space the semigroup constructed by Miller [13] can be considered the adjoint semigroup of the semigroup constructed by Barbu and Grossman [2]. This provides a useful characterization of Miller’s semigroup which can be applied to obtain additional information about the semigroup defined by Barbu and Grossman.
33 citations
TL;DR: McAlister's Theorem A and B as discussed by the authors states that every E-unitary inverse semigroup is isomorphic to a quotient of a P-semigroup by an idempotent-separating congruence.
Abstract: By an E-unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest.An important structure theorem for E-unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G, a partially ordered set and a subset , satisfying certain conditions, he constructs an E-unitary inverse semigroup P(G, ). A semigroup of this type is called a P-semigroup. The structure theorem states that every E-unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlister's Theorems A and B respectively. A triple (C, ) of the type used to construct a P-semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E-unitary inverse semigroup.
23 citations
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18 citations
01 Jan 1976
TL;DR: In this paper, the authors give a short and simple proof to a more general version of a recent result of Yeadon for semigroups of weak*-continuous operators on a dual Banach space.
Abstract: In this paper, we give a short and simple proof to a more general version of a recent result of Yeadon for semigroups of weak*-continuous operators on a dual Banach space. Our result has application to amenable groups and property P of a von Neumann algebra.
17 citations
16 citations
TL;DR: In this article, it was shown that the translational hull of a topological semigroup can be computed in terms of the topology of continuous convergence and the compact-open topology.
Abstract: This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull Sl(S) is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact- open topology in this case), then Sl(S) is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divis- ibility are presented. Techniques for determining the translational hull of cer- tain types of topological semigroups, along with numerous examples, are set forth in the final section.
TL;DR: The bicyclic semigroup has been shown to have a unique non-trivial closure obtained by adjoining the integers Z as discussed by the authors, which is the closure of the bicyclic topological inverse semigroup.
Abstract: The structure of a bisimple w-semigroup has been shown by Reilly [5] to be entirely determined by its group of units G and an endomorphism ~ of G. The bicyclic semigroup, when considered as a topological subsemigroup of a locally compact topological inverse semigroup, has been shown by Eberhart and Selden [2] to have a unique non-trivial closure obtained by adjoining the integers Z. We obtain a pair of theorems. The first shows that given any bisimple w-semigroup 6 with G compact and ~ onto, a certain construction produces a locally compact topological inverse semigroup which is the closure of ~. The second shows that the locally compact closure of every bisimple w-semigroup with G compact and ~ onto is algebraically and topologically isomorphic to one of these constructions. I. 8ISIMPLE ~-SEMIGROUPS IN THE LOCALLY COMPACT SETTING. In a semigroup the set E of idempotents can be partially ordered by defining e ~ f if and only if ef = fe e. A semigroup is called an w-semigroup if
TL;DR: In this paper, it was shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors F:X -* G, where X is a compact semilattice and G is the category for compact groups and continuous homomorphisms.
Abstract: A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors F:X -* G, where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from F:X -* G to G:Y -* G is a pair (e, w) such that e is a continuous homomorphism of X into Y and w is a natural transformation from F to Ge. Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings. A popular topic of study in compact semigroups has been the question, for a given compact Hausdorff space, how many nonisomorphic continuous, associative multiplications of a given type will it admit? There is an older companion question, and that is, for a given algebraic structure, is there a compact Hausdorff topology which is compatible with all the operations, and if so, how many such topologies exist? It is known that the abelian groups which admit such a compact Hausdorff topology are certain products of copies of the group of rational numbers, p-adic groups, finite groups, and Z(p°°) [4, Theorem 25.25]. Butan abelian group may admit several such topologies. For example, the additive group of real numbers admits a compact /i-dimensional topology for each positive integer n. In the nonabelian case, there is the 1932 result by van der Waerden [9], in which he described a system of \"neighborhoods\" about the identity of any group, which was finer than any compact group topology for which the identity was not isolated in the set of noncentral elements. He further proved that if a group admitted a topology giving it the structure of a compact simple Lie group, then each of these algebraically defined neighborhoods was a neighborhood of the identity relative to the given topology. Thus he gave what amounted to an algebraic description of the topology of a compact simple Lie group, which had an immediate Received by the editors November 1, 1974. AMS (MOS) subject classifications (1970). Primary 22A1S.
TL;DR: The semigroup of all one-to-one mappings of a set to itself is shown to have the property that every automorphism is inner as discussed by the authors, which is the same as the semigroup for all automorphisms.
Abstract: The semigroup of all onto mappings of a set to itself and the semigroup of all one-to-one mappings of a set to itself are shown to have the property that every automorphism is inner.
TL;DR: In this article, the concept of invariance of a ring relative to a commutative semigroup was studied and proved for certain Prufer rings and affine algebras relative to semigroups of a special class.
Abstract: In this paper we study the concept of the invariance of a ring relative to a commutative semigroup. Invariance is proved for certain Prufer rings and affine algebras relative to semigroups of a special class.Bibliography: 11 titles.
TL;DR: In this paper, a Galois correspondence between the subsemigroups of TX and a particular class of digraphs on X is established. But the correspondence is restricted to the case where the digraph is a proper subsemigroup of S * 9 (S).
Abstract: Let Tx be the full transformation semigroup on the set X and let S be a subsemigroup of TX. We may associate with S a digraph g (S) with X as set of vertices as follows: I -k j E g(S) iff there exists at E S such that a(i) = j. Conversely, for a digraph G having certain properties we may assign a semigroup structure, S(G), to the underlying set of G. We are thus able to establish a "Galois correspondence" between the subsemigroups of TX and a particular class of digraphs on X. In general, S is a proper subsemigroup of S * 9 (S).
TL;DR: This paper constructs for a given algebra automaton the semigroup of functions which can be defined by that algebra Automaton as an operator semigroup with prescribed generators and characterize the algebra automata of algebras in a particular variety.
TL;DR: In this paper, the authors deal with time evolution of a finite quantum system given by a dynamical semigroup Λ t, and define and give some properties of the convex subset of states "pathological" in some aspect evolving in strictly reversible manner independently of the stochastic surroundings of the system.
01 Jan 1976
TL;DR: The Groupe d'étude d'algèbre as discussed by the authors, 1975-1976, tous droits réservés, implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Groupe d’étude d’algèbre (Secrétariat mathématique, Paris), 1975-1976, tous droits réservés. L’accès aux archives de la collection « Groupe d’étude d’algèbre » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: The boundary of a subset A in S is the set OA = ~i A S\ A, where the bar denotes closure as discussed by the authors, and the boundary of A is defined as the set R(A) ~ (xES : x'EA for some positive integer n).
Abstract: In what follows S is always a topological semigroup, i.e., S is an algebraic semigroup and also a Hausdorff space such that the multiplication is continuous. The boundary of a subset A in S is the set OA = ~i A S\ A, where the bar denotes closure. S~uM considered in [3] the boundaries of radicals (see below for definition) of ideals in S; it is the purpose of this note to obtain some further results about boundaries in semigroups. For convenience, we shall assume the boundaries under consideration are non:empty sets. Given non-empty subsets A, B in S, we say that A is a relative ideal of B if ABcA and BAcA (cf. [1]); if, in addition, AcB and B is a semigroup, then A will be an ideal of the semigroup B in the ordinary sense. T~EOREM 1. Let A be an open ideal of S. Then OA is a relative ideal of S\A if and only if S'~A is a semigroup. In such case, ~A is an (ordinary) ideal of the semigroup S~A. P~OOF. Note that 0A ~-z/(7 (S~,A). Suppose S~A is a semigroup, and let xEOA, yES'~A; then xyE~ and xyES~A, whence xyEOA. It follows that OA , (5"~,~A)cOA, and similarly (S'~.A)OAcOA as required. Conversely, if ~A is a relative ideal of S~A, take x, yES~A. Then OA �9 xc cOA and yoAcOA, implying that ~A.xy~Ac(OA)2"cO A.(S\A)c OAcS\A. In view of the fact that A is an ideal, we see that xyES'~A. Therefore, S\A is a semigroup, completing the proof. :For a non-empty subset A c S, we define the radical of A as the set R(A) ~ (xES : x'EA for some positive integer n}. Note that, if A is open, R(A) is open (by Lemma A of [3], or following from the fact that R(A) ~-- U f~i(A) where fn :S-+ S is defined byfn(X ) = x n for xES).