Showing papers on "Cancellative semigroup published in 1991"

Higher order relations for a numerical semigroup

Abstract: © Université Bordeaux 1, 1991, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). 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.

Perturbation of positive semigroups

Abstract: The purpose of this note is to study perturbations of generators of positive semigroups by positive operators. Let E be a complex Banach lattice and A be a linear operator on E with domain D (A). We say that A is resolvent positive if there exists w e P~ such that (2 - A): D (A) --~ E is bijective and (2 - A)- 1 is a positive operator on E for all 2 > w. Note that the generator of a positive semigroup is resolvent positive. Assume that A generates a positive semigroup (by which we always mean a C0-semi- group) and B: D(A)---* E is linear and positive such that A + B (with domain D (A + B) = D (A)) is resolvent positive. Then it was shown by Desch [8] that A + B generates a positive semigroup whenever E is a space L 1. A simple proof is given by Voigt [20]. If E is an LP-space, 1 < p < 0% then the assertion is false, in general (see [4]). However, we show in Section 1 that in the case where the semigroup generated by A is holomorphic, also A + B generates a holomorphic semigroup without any restriction on the space. Furthermore, we prove in Section 2 that A + B generates a semigroup whenever B is a positive rank-one perturbation of A. This is remarkable in view of a recent result of Desch-Schappacher [9]. If the semigroup generated by A is not holomorphic, there always exists a (necessarily non positive) rank-one perturbation B such that A + B is not a generator. In Section 3 we give a criterion for perturbation by multiplication operators which, in view of the Sobolev embedding theorems, is particularly useful for elliptic operators. As an illustrating example we consider Schr6dinger operators. In Section 4 the results are applied to systems of evolution equations, which obtained special attention recently (see [14]). Concerning terminology and basic results we follow [17] and [13]. A c k n o w 1 e d g e m e n t. We are indebted to J. Voigt for several valuable suggestions and comments. 1.

Normal semigroups of one-to-one transformations

Abstract: If a transformation semigroup S is defined by means of certain properties there is aproblem of determining the elements of S explicitly. In this paper, the above problem isconsidered for ^-normal semigroups S of total one-to-one transformations of a set X.A transformation semigroup S on X is termed ^-norma

Necessary and sufficient conditions for conservativeness of dynamical semigroups

Abstract: Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let ℋ be a Hilbert space andA a von Neumann algebra. A dynamical semigroup Pt is a σ-weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup Pt possessing the property of preserving the identityI∈A is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA+. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra ofℒ∞(Rn) yields a series of new regularity conditions for both diffusion and jump processes.

On essential right congruences of a semigroup

Abstract: In ring theory one can give several approaches to the introduction of the concept of semisimplicity and a large number of equivalent formulations of this concept [5, 13]. Analogues of some of these formulations have been made, and studied, for semigroups [4, 6, 7, 8, 10, 11, 12] in terms of ideals or congruence relations. It seems possible that suitable and effective analogues can be made for each of these ring theoretical formulations in terms of congruence relations. However, unlike the situation for rings most of these analogues give inequivalent formulations in semigroups. One of the weakest of these is the nonexistence of a proper, essential right congruence. A right congruence ~ in a semigroup S is essential if for any right congruence a we have Q N a = z (the identity relation) implies a = z. The main result of this paper is a characterization of semigroups with the d.c.c, on right ideals and having no proper essential right ideals and having no proper essential right congruences. The first step in this characterization is a description of the lattice of right ideals in such a semigroup. Our results of this description in Section 2 should be compared with the work of Feller and Gantos [2] and Fountain [3]. While the class of semigroups studied in these two papers are defined quite differently than the class in this paper there is a striking similarity in the results; thus, suggesting a common area of investigation. In the subsequent sections the main technique used is the examination of a selection of proper right congruences and the implications on the multiplicative properties obtained from the assumption of nonessentiality.

Semigroup compactifications by generalized distal functions and a fixed point theorem

Abstract: The notion of “Semigroup compactification” which is in a sense, a generalization of the classical Bohr (almost periodic) compactification of the usual additive reals R , has been studied by J. F. Berglund et. al. [2]. Their approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of commutative C * algebras, where the spectra of admissible C * -algebras, are the semigroup compactifications. H. D. Junghenn's extensive study of distal functions is from the point of view of semigroup compactifications [5]. In this paper, extending Junghenn's work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroup S , and show that these function spaces are admissible C * - subalgebras of C ( S ) . We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactifications enjoy. In our work, as it is in Junghenn's, the Ellis semigroup plays an important role. Also, relating the existence of left invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem.

Solvability and the permutation property in inverse semigroups

Abstract: If any product of three elements of an inverse semigroup S can be re-ordered, then S is solvable; the same is not true for any integer number greater than three.

Universal groups on semilattices of reversible cancellative semigroups

Abstract: LetS be a semigroup which is a semilattice Ω of reversible cancellative semigroupsS α, α∈Ω. This paper studies the relationship between the universal groupG onS and the universal groupsG α onS α. We also show that the universal homorphismsf α∶S α→G α, α∈Ω fromS α to the category of groups combine to a homomorphismf∶S→G ofS into the category of groups.