On the E-unitary covers for a Bruck semigroup

We give a characterization of theE-unitary covers for a Bruck semigroup, which is a generalization of Theorem 3 given in [1]. In a recent paper [1] we gave a characterization of theE-unitary covers for a Bruck semigroupB(T,α) whereT is a finite chain of groups, and now we give a characterization forB(T,α) whereT is any inverse semigroup. We use here the notations and the terminology of Petrich's book [2]. First we prove a Theorem which is more general than [[1], Theorem 2]. I wish to express my thanks to Dr. G. Pollák for his valuable advice.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

On the construction of the conjugate hull of Brandt semigroups

In this paper the characterizations of conjugate hulls ψ (S). ϕ(S),T(S) and θ(S) on a Brandt semigroupS are given. By using these results we can pro thatT(S) is self-conjugate in ψ(S) for a Brandt semigroupS. This work is supported by National Natural Science Foundation of China     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS * _U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgamS * _U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv<A ∪B :u =v > whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A ⁻¹ (resp.B ∪B ⁻¹) isE-unitary. Research of all authors supported by a grant from the Italian CNR. The first and third authors’ research was partially supported by MURST. The second author’s research was also partially supported by NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln.     

Range sets and partition sets in connection with congruences and algebraic invariants

With a semigroup of transformationsS, we associate a class of equivalence relations onR(S) (calledclosed under inclusion relations), the set of ranges ofS. We define a new notion of connectedness for semigroups of transformations (calledrange-connectedness). For a range-connectedS, the closed under inclusion relations and the left-zero congruences ofS are dually isomorphic. The ideas above are dualized for the partition sets ofS. We associate withS an ordered pair which measures its range and partition connectedness. We generalize to an arbitrary semigroupT by considering faithful representations ofT by semigroups of transformations. In so doing, we are able to define an algebraic invariant for semigroups.     

Necessary and sufficient conditions for stability of effectivity functions

An effectivity functionE assigns to every coalitionS of players a familyE (S) of subsetsB of an outcome setA such thatS can force the outcome to belong to any of the setsB inE (S). The effectivity functionE is stable if for every preference profile there is an outcomex with the property that there is no coalitionS and subsetB ofA such thatB εE (S) and each player inS prefers everyy εB tox. The paper gives a necessary and sufficient condition for an effectivity function to be stable.     

The structure of 0-bisimple strongly E *-unitary inverse monoids

We introduce a class of strongly E ^*-unitary inverse semigroups S _i (G, P) (i = 1,2) determined by a group G and a submonoid P of G and give an embedding theorem for S _i (G, P). Moreover we characterize 0-bisimple strongly E ^*-unitary inverse monoids and 0-bisimple strongly F ^*-inverse monoids by using S _i (G, P).     

Su una condizione di finitezza per semigruppi finitamente generati

We prove that a finitely generated semigroupS is finite if and only if there exists an integern such that in each sequence ofn elements ofS there exist two different non empty factors with the same value inS. We prove this result using only elementary facts concerning the canonical form of an element of a finitely generated semigroup.     

Vector representable matroids of given rank with given automorphism group

A simple way of associating a matroid of prescribed rank with a graph is shown. The matroids so constructed are representable over any sufficiency large field. Their use is demonstrated by the following result: Given an integer k?3 and a function G associating a group with each subset of a set S, there is a matroid M(E), representable over any sufficiently large field, such that E ? S, and for any T ?/ S, the rank of M/Tis k, and the automorphine group of MT is isomorphic to G(T).     

A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

Certaines formes linéaires pathologiques sur un espace de Banach dual

We construct a Banach spaceE such thatE′ isw ^*-separable, andf∈E″/E, which isw ^*-continuous on every set ofE′ which is thew ^*-closure of a countablebounded set ofE′.      

Factor-powers of finite symmetric groups

To a transformation semigroup (S, M) we assign a new semigroupFP(S) called the factor-power of the semigroup (S, M). Then we apply this construction to the symmetric groupS _n. Some combinatorial properties of the semigroupFP(S _n) are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of ordern and the structure of its idempotents. The idempotents are used in characterizingFP(S _n) as an extremal subsemigroup of the semigroupB _n of all binary relations of ann-element set and also in the proof of the fact thatFP(S _n) contains almost all elements ofB _n.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 176–188, August, 1995.This work was partially supported by the Foundation for Basic Research of the Ukrainian State Committee on Science and Technology.     

Automorphisms of linear semigroups over division rings

Given an-dimensional right vector spaceV over a division ring we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf ⁻¹, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.     

E ∨-unitary covers for ∨-semilatticed inverse semigroups

It is well known that every inverse semigroup admits an E-unitary cover. In this paper we investigate the analogue of E-unitary covers within the variety of ∨-semilatticed inverse semigroups.     

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 set intersection theorem and applications

Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM ₁ ? M _n be closed sets inR ⁿ which cover the standard simplexS, so thatM _i coversS _i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM _i andk-almost-completely labeled if it belongs to all butM _k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point. This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR ⁿ . It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.