Showing papers on "Bicyclic semigroup published in 1993"
TL;DR: A semilattice of simple poe -semigroups is defined in this article as an ordered semigroup having a greatest element, that is, an ordered order semigroup (:po -semigroup) having the greatest element.
Abstract: -semigroup -that is an ordered semigroup (:po -semigroup) having a greatest element - is a semilattice of simple semigroups if and only if it is a semilattice of simple poe -semigroups [3].
50 citations
01 Oct 1993
TL;DR: In this article, a correspondence between a class of coverings of an inverse semigroup S and embeddings of S was established, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.
Abstract: A correspondence is established between a class of coverings of an inverse semigroup S and a class of embeddings of S, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.
14 citations
01 Jan 1993
TL;DR: In this paper, basic system theoretic concepts for abstract systems of the form of strongly continuous semigroup S(t) on a Banach space Z and necessary and sufficient conditions for this to be the case are given by the Hille-Yosida theorem are introduced.
Abstract: In this paper some basic system theoretic concepts will be introduced for abstract systems of the form
$$\dot x\left( t \right) = Ax\left( t \right) + Bu\left( t \right), x\left( 0 \right) = x^0 , y\left( t \right) = Cx\left( t \right)$$
(1)
Here A is the infinitesimal generator of a strongly continuous semigroup S(t) on a Banach space Z and necessary and sufficient conditions for this to be the case are given by the Hille-Yosida theorem. For U another Banach space B ∈ L(U, Z) and x0 ∈ Z, u(·) ∈ L2(0, ∞; U) a mild solution is defined to be
$$x\left( t \right) = S\left( t \right)x^0 + \smallint _0^t S\left( {t - s} \right)Bu\left( s \right)ds$$
(2)
and x(·) ∈ C(0; ∞; Z). Various definitions of controllablity, observability, stabilizability, detectability, identifiability and realizability will be given and theorems which characterize them will be stated. Throughout the paper examples will be given (albeit trivial ones) which illustrate the way the abstract definitions and results can be applied to concrete problems defined via partial differential equations and delay equations.
12 citations
TL;DR: A semigroup with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S, (3t G T)(Vu, v 6 T) tt* = uv' => u = v. as discussed by the authors.
Abstract: A semigroup 5 with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S, (3t G T)(Vu, v 6 T) tt* = uv' => u = v. It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive. 1. DEFINITIONS
12 citations
TL;DR: In this paper, a simple solution of the word problem in the free combinatorial strict inverse semigroup is presented, where the associated partial order is realized as a set of symmetric and transitive relations on a certain set and the corresponding Brandt semigroups as well as the partial homomorphisms can be obtained in a canonical way.
9 citations
TL;DR: The semigroup End(X, ⩽) of isotonic mappings of a linearly ordered set as well as the transition semigroups of automata that arise from certain varieties of formal languages are discussed.
9 citations
5 citations
TL;DR: In this article, the authors define the Andre]Quillen cohomology T s T A, A; k n G 0, A. The first three modules are important for the deformation theory of A or its geometric equivalent Spec A: T 1 equals the set of infinitesimal deformaA tions, T 0 describes their automorphisms, and T 2 contains the obstructions A A for lifting infiniteimal deformations to larger base spaces.
4 citations
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TL;DR: In this article, a spectral mapping theorem for continuous semigroups of operators on any Banach space is presented, where the condition for the hyperbolicity of a semigroup on the Banach spaces is given in terms of the generator of an evolutionary semigroup acting in the space of functions.
Abstract: We present a spectral mapping theorem for continuous semigroups of operators on any Banach space $E$. The condition for the hyperbolicity of a semigroup on $E$ is given in terms of the generator of an evolutionary semigroup acting in the space of $E$-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in $E$ is also studied. A ``discrete'' technique for the investigating of the evolutionary semigroup is developed and applied to describe the hyperbolicity (exponential dichotomy) of the nonautonomuos equation.
File Length: 68K
3 citations
3 citations
TL;DR: In this article, it was shown that a completely regular semigroup is in the semigroup variety generated by the bicyclic semigroup if and only if it is an orthogroup whose maximal subgroups are abelian.
Abstract: We shall show that a completely regular semigroup is in the semigroup variety generated by the bicyclic semigroup if and only if it is an orthogroup whose maximal subgroups are abelian. Therefore the lattice of subvarieties of the variety generated by the bicyclic semigroup contains as a sublattice a countably infinite distributive lattice of semigroup varieties, each of which consists of orthogroups with maximal subgroups that are torsion abelian groups. In particular, every band divides a power of the bicyclic semigroup.
01 Mar 1993
TL;DR: In this article, it was shown that the natural rep- resentation of a small semigroup has a finite basis of identical relations and discussed this fact in a general context of universal algebra.
Abstract: An equational theory of a very small semigroup may fail to be finitely presented. A well-known example of such a semigroup was studied in detail by Peter Perkins some twenty years ago. We prove that the natural rep- resentation of his semigroup has a finite basis of identical relations and discuss this fact in a general context of universal algebra.
TL;DR: In particular, if a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for any A∈V and any left cancellative monoidM, there is a semigroupS ∈ V such that A is a retract of S and M is isomorphic to the monoid of all injective endomorphisms of S as mentioned in this paper.
Abstract: If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.
01 Jan 1993
TL;DR: In this article, the second adjoint of a Co-semigroup of linear operators on a Banach space X is shown to be strongly continuous for t > O, resp. t > 0.
Abstract: Let T(t) be a Co-semigroup of linear operators on a Banach space X, and let X@, resp. X( , denote the closed subspaces of X* consisting of all functionals x* such that the map t 4 T*(t)x* is strongly continuous for t > O, resp. t > O. Theorem. Every nonzero orbit of the quotient semigroup on X*/X? is nonseparably valued. In particular, orbits in X*/XO are either zero for t > 0 or nonseparable. It also follows that the quotient space X* /X? is either zero or nonseparable. If T(t) extends to a Co-group, then X*I/X is either zero or nonseparable. For the proofs we make a detailed study of the second adjoint of a Cosemigroup.