Powers of subsets in a finite semigroup

Communicated by Boris M. Schein     

Balanced subsets of a symmetric inverse semigroup

For elements in the subsets of some symmetric inverse semigroup we study the problem of equal cardinality for the sets of commuting and noncommuting elements.     

Symmetry classes of tensors via semigroup operands

The symmetry of tensors, such as the symmetric or antisymmetric ones (built on a finite-dimensional complex vector space) may be described by a complex-valued homomorphism of the symmetric group with the specification that its action equal scalar multiplication by the value, e.g. by 1 or sign. This condition may be construed as a universalizing operand (over the symmetric group with 0) homomorphism from the unsymmetrized tensors—a restructuring which permits a clearer and more effective treatment of these symmetries; freed from the multilinear setting in which they arose, it also points the way to a development of semigroup symmetries on more general universal algebras.     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

Centers of semigroup rings and conjugacy classes

Communicated by Boris M. Schein     

Balanced subsets of the semigroup of partial transformations

For the elements of subsets of the semigroup of partial transformations we study when the sets of commuting and noncommuting elements are of the same cardinality.     

Radicals of some semigroup algebras

In this paper we seek to determine the Jacobson radical of certain algebras based on semigroups, and in particular on the semigroups (βS,□), where S is a cancellative, countable, abelian semigroup and βS is its Stone–?ech semigroup compactification. In particular, we wish to determine the radical of ? ^?1(β?).     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

One-sided identities for subsets of the semigroup of binary relations

We describe one-sided identity elements and subsets of the semigroup of all binary relations on a nonempty set. We obtain formulas for the number of identities for a finite set.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1026–1031, August, 1990.     

One-sided zeros of subsets of the semigroup of binary relations

The one-sided zeros of the elements and of the subsets of the semigroup of all binary relations on an arbitrary nonempty set, are described. For a finite set of zeros, formulas are obtained for their number.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 600–604, May, 1990.     

The large rank of a finite semigroup using prime subsets

The large rank of a finite semigroup \(\Gamma \) , denoted by \(r_5(\Gamma )\) , is the least number \(n\) such that every subset of \(\Gamma \) with \(n\) elements generates \(\Gamma \) . Howie and Ribeiro showed that \(r_5(\Gamma ) = |V| + 1\) , where \(V\) is a largest proper subsemigroup of \(\Gamma \) . This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.     

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.     

Dense subsets of the boundary of a Coxeter system

In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system , if is quasi-dense in and the order for some , then there exists a point in the boundary of the Coxeter system such that the orbit is dense in . Here . We also show that if the set is quasi-dense in , then is dense in .

     

Quiver approach to some ordered semigroup algebras

The aim of this paper is to research the relation among generalized path algebras, pseudo-admissible ordered semigroup algebras and path algebras over algebras. First, the Gabriel theorem for contract pseudo-admissible ordered semigroup algebras is given. Second, a family of pseudo-admissible ordered semigroups is mixed together via the method of generalized path algebras to construct a new pseudo-admissible ordered semigroup as the extended version. Finally, we characterize the path algebra over an algebra in two ways through normal and non-normal generalized path algebras, respectively, over a field.     

The groups of the semigroup of compact subsets of a topological group

The compact subsets of a topological groupG form a semigroup,S(G), when multiplication is defined by set product. This semigroup is a topological semigroup when given the Vietoris topology. It would be expected that the subgroups ofS(G) should in some way be related to the groupG. This is the case. It is shown that the subgroups ofS(G) are both algebraically and topologically exactly the groups obtained as quotients of certain subgroups ofG. One consequence of this is that every subgroup ofS(G) is a topological group. Conditions are also given for these subgroups to be open or closed. Green's relations inS(G) have a particularly nice formulation. As a result, the relationsD andJ are equal inS(G). Moreover, the Schützenberger group of aD-class is a topological group that is topologically isomorphic to a quotient of certain subgroups ofG.