Subnormal semigroups of composition operators

In this article we describe a model for subnormal semigroups of composition operators (with linear fractional symbol) acting on the Hardy space . We also discuss cyclicity of such semigroups in the context of more general results studied by J. H. Shapiro and P. S. Bourdon.

     

Duals and topological center of a class of matrix algebras with applications

We characterize the topological center of a class of matrix algebras, which are called -Munn algebras. This involves a characterization of Arens regular semigroup algebras for a large class of semigroups, which is an extension of Young's Theorem for semigroups. We show by some counter examples that only up to a certain extent Young's Theorem can be generalized.

     

Product systems over right-angled Artin semigroups

We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid . We show that the existing notions of product systems fit into our categorical framework, as do the -graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid . In particular, we obtain necessary and sufficient conditions under which a collection of -graphs form the coordinate graphs of a -graph.     

Semigroups for which every S-system has the natural partial order

In this note we characterize semigroups S having the property that the relation , defined by x y iff x yS U {y}, partially orders every S-system.Research of second author supported in part by a post-doctoral fellowship in the Biomathematics Program at North Carolina State University under PHS Grant # GM -678 from NIGMS and while on leave from Bowling Green State University.TO ALFRED H. CLIFFORD on his 65th birthday on the 11th of July, 1973     

On <Emphasis Type="Italic">n</Emphasis>-Central Semigroups

A semigroup S is said to be n-central if xⁿ belongs to the center of S for every x S. We prove that every n-central semigroup is a semilattice of archimedean n-central semigroups. We obtain characterizations of simple (0-simple) n-central semigroups and describe subdirectly irreducible n-central semigroups. We also deal with the connection of n-central semigroups and E-k semigroups.     

Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

GO-spaces with -closed discrete dense subsets

In this paper we study the question ``When does a perfect generalized ordered space have a -closed-discrete dense subset?' and we characterize such spaces in terms of their subspace structure, -mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a -closed-discrete dense set and a weak monotone ortho-base. That metrization theorem cannot be proved in ZFC for perfect GO-spaces because if there is a Souslin line, then there is a non-metrizable, perfect, linearly ordered topological space that has a weak monotone ortho-base.

     

On Aperiodic Relational Morphisms

We characterize aperiodic relational morphisms as those that are injective on regular $\H$-classes. This result is applied to obtain simple proofs and generalizations of McAlisters results on joins of aperiodic semigroups and groups. Also, we show that if $\pv H$ is a proper, non-trivial pseudovariety of groups, then \[\pv A\ast \pv H\subsetneq (\pv A\ast \pv G)\cap \ov {\pv H}.\] We provide coordinate-free formulations and proofs of Rhodess Presentation Lemma and generalizations. As an application, we give simpler proofs of Tilsons theorem on the complexity of semigroups with at most $2$ non-zero $\J$-classes and Rhodess theorem that complexity is not local.     

Semilattices of proper inverse semigroups

Given a semilattice Y of inverse semigroups S, there corresponds a semilattice Y of groups G in a natural way. This correspondence is used to study semilattices of proper inverse semigroups. In paticular, it is shown that if S is a semilattice of proper inverse semigroups, then there exists a minimum semilattice congruence such that each -class is proper and there exists a maximum semilattice congruence such that each -class is proper.     

The Universal Covering of an Inverse Semigroup

We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos . In particular, we prove that the fundamental group of coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.     

A Lattice of Quasivarieties of Normal Cryptogroups

A normal cryptogroup S is a completely regular semigroup in which is a congruence and is a normal band. We represent S as a strong semilattice of completely simple semigroups, and may set For each we set and represent by means of an h-quintuple These parameters are used to characterize certain quasivarieties of normal cryptogroups. Specifically, we construct the lattice of quasivarieties generated by the (quasi)varieties and This is the lattice generated by the lattice of quasivarieties of normal bands, groups and completely simple semigroups. We also determine the B-relation on the lattice of all quasivarieties of normal cryptogroups. Each quasivariety studied is characterized in several ways.     

Zur Konstruktion von weierstrasspunkten mit Vorgegebener Halbgruppe

Let H be a semigroup generated by 4 elements such that k[H] is an almost complete intersection. Then there is a Riemann surface and a Weierstraß point on it with H as semigroup of nongaps. In order to prove this, a structure theorem for such semigroups is given.     

The Dual of Semigroup Algebras with Certain Locally Convex Topologies

In this paper, we deal with the dual of the semigroup algebras for an extensive class of locally compact semigroups S under certain locally convex topologies. We first introduce and study a locally convex topology on under which the Banach space can be identified with its strong dual. We then show that, except for the case where S is finite, there are infinitely many such locally convex topologies on . Finally, we characterize the spectrum of in terms of semicharacters on S.     

Long-term stability of variable stepsize approximations of semigroups

This paper is concerned with the stability of rational one-step approximations of semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Under additional hypotheses on the approximation, more favorable stability bounds are obtained for the subclass of holomorphic semigroups.

     

Perfect equilibria of a model of n-person noncooperative bargaining

We study the set of subgame perfect equilibria associated with then-person noncooperative bargaining mechanism proposed by Hart and Mas-Colell (1992). Our results pertain to transferable utility games. The set of perfect equilibria depends on the parameter representing the continuation probability, . For general TU games, we characterize the set of payoffs from perfect equilibria for (a) small values of and (b) large values of . For symmetric games a complete characterization for all values of is provided.We are grateful to Andreu Mas-Colell for help and encouragement and to two referees for every helpful comments.     

A Hille-Yosida theorem for Bi-continuous semigroups

In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous, we introduce the concept of bi-continuous semigroups defined on Banach spaces with an additional locally convex topology . On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous for and such that the orbit maps are -continuous. We then apply the result to semigroups induced by flows on a metric space as studied by J. R. Dorroh and J. W. Neuberger.     

Regular semigroups: amalgamation and the lattice of existence varieties

A class of regular semigroups is called an existence variety (ore-variety) if it is closed under , and , the operations of taking all morphic images, all regular subsemigroups, and all direct products, respectively; and such classes were previously shown (like varieties) to be determined by sets of identities. We determine whiche-varieties of regular semigroups have the amalgamation property. Preliminary results include (1) a construction of the lattice ofe-varieties of generalized inverse semigroups and (2) the result that thee-variety of combinatorial, strict, regular semigroups is the leaste-variety which is notE-solid.Dedicated to Professor G. B. Preston, on the occasion of his 65th birthdayPresented by Ralph Freese.     

A Representation for $F$-Regular Semigroups

A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence _S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/_S, where X belongs to the band variety, generated by the band of idempotents E_S of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case.     

Semigroups whose Proper Right Congruences Form a Tree

Let S be a semigroup whose set of proper right congruences form a tree. The main theorem is a characterization of those semigroups having this property. In this characterization we draw on the results of Schein and Tamura for commutative semigroups and Kozhukhov for left chain semigroups and Hitzel for nilpotent semigroups. The interested reader should also see the work of Nagy on -semigroups.