Regular F-Abundant Semigroups

ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

A complete theory for jointly continuous nonlinear semigroups on a complete separable metric space

For a jointly continuous semigroup of transformations on a complete separable metric space X an induced semigroup of linear transformations on an appropriate space of measures is defined. A complete characterization of generators of such semigroups is given and it is shown how to construct a jointly continuous semigroup on X from a generator taken from this characterized collection.     

The idempotent-separating degree of a block-group

In this note we characterize the least positive integer n such that there exists an idempotent-separating homomorphism from a finite block-group S into the monoid of all partial transformations of a set with n elements. In particular, as for a fundamental semigroup S this number coincides with the smallest size of a set for which S can be faithfully represented by partial transformations, we obtain a generalization of Easdown’s result established for fundamental finite inverse semigroups. The author gratefully acknowledges support of FCT and FEDER, within the project POCTI-ISFL-1-143 of CAUL, and the fellowship SFRH/BSAB/244/2001.     

Normally Ordered Inverse Semigroups

NO of all normally ordered inverse semigroups. We show that the pseudovariety of inverse semigroups PCS generated by all semigroups of injective and order partial transformations on a finite chain consists of all aperiodic elements of NO . Also, we prove that NO is the join pseudovariety of inverse semigroups. PCS V G , where G is the pseudovariety of all finite groups.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

The Congruence Lattice of Bruck–Reilly Extensions

We study the lattice. C(S) of congruences of a monoid S which is the Bruck-Reilly extension of a monoid T by a homomorphism . The inclusion, meet and join of congruences are described in terms of congruences and ideals of T. We show that C(S) can be naturally decomposed into three sublattices, corresponding (roughly speaking) to the three different types of congruences on such semigroups.1991 Mathematics Subject Classification: 20M10     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

Retracts and monomorphism monoids in semigroup varieties

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.     

Inverse semigroups and varieties of finite semigroups

In this paper we shall present a new construction of the free inverse monoid on a set X. Contrary to the previous constructions of [9, 11], our construction is symmetric and originates from classical ideas of language theory. The ingredients of this construction are the free group on X and the relation that associates to a word w of the free monoid on X, the set of all pairs (u, v) such that uv = w. It follows at once from our construction that the free inverse monoid on X can be naturally embedded into the Schützenberger product of two free groups of basis X. We shall also give some connections with the theory of expansions as developed by Rhodes and Birget [2, 3].     

Sulľapplicazione della funzione ellitticap u alla teoria dei poligoni di poncelet

Abstract  In this paper we study strongly continuous positive semigroups on particular classes of weighted continuous function space on a locally compact Hausdorff space X having a countable base. In particular we characterize those positive semigroups which are the transition semigroups of suitable Markov processes. Some applications are also discussed. Keywords Positive semigroup, Markov transition function, Markov process, Weighted continuous function space, Degenerate second order differential operator Mathematics Subject Classification (2000) 47D06, 47D07, 60J60     

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed.

     

Small inherently nonfinitely based finite semigroups

We show how to construct all ``forbidden divisors' for the pseudovariety of not inherently nonfinitely based finite semigroups. Several other results concerning finite semigroups that generate an inherently nonfinitely based variety that is miminal amongst those generated by finite semigroups are obtained along the way. For example, aside from the variety generated by the well known six element Brandt monoid \tb , a variety of this type is necessarily generated by a semigroup with at least 56 elements (all such semigroups with 56 elements are described by the main result). September 23, 1999     

The word problem for nilpotent inverse monoids

The concept ofk-nilpotency is adapted for inverse semigroups andN _k(X) is defined as the largestk-nilpotent Rees quotient of the free inverse monoid onX. The membership problem is solved for a certain class of ideals of quasifree inverse monoids. As a consequence, the word problem is shown to be decidable for every finite relation onN _k (X), producing an unusual example of total decidability. The author acknowledges support from JNICT-Project SAL (PBIC/C/CEN/1021/92) and ESPRIT-BRA Working Group 6317 “ASMICS”.     

A Note on Prehomomorphisms

We show that if G is a free group with basis X then any map θ from X to an inverse monoid S extends to a monoid prehomomorphism ψ: G\rightarrow S. As an application we give an affirmative answer to a problem of M. Petrich. 1980 Mathematics Subject Classification: Primary 20M10. September 14, 1999     

Clifford semigroups as functors and their cohomology

We prove that the category of Clifford semigroups and prehomomorphisms CSP\mathcal{CSP} is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we find a very close relationship between the cohomology groups of a commutative inverse monoid and the cohomology groups of the colimit group of the functor giving the monoid.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

w-Hyponormal Operators are Subscalar

We prove that if X is a Banach space, R, S B(X), then RS is subscalar (subdecomposable) if and only if SR is. As corollaries, it is shown that w-hyponormal operators (including p-hyponormal (p > 0) and log-hyponormal operators) and their Aluthge transformations and inverse Aluthge transformations are subscalar.