Regular elements in sandwich semigroups of binary relations

This note gives a counterexample to a published characterization of regular elements in sandwich semigroups of binary relations. The method of that paper is used to characterize those elements having a right identity. A characterization of regular elements is obtained following the approach of Schein.     

On generating regular elements in the semigroup of binary relations

We prove the semigroup generated by four binary relations contains all regular binary relations.     

Sandwich semigroups of binary relations

This paper introduces new semigroups of binary relations that arose naturally from investigating the transfer of information between automata and semigroups associated with automata. In particular we introduce a new multiplication on binary relations by means of an arbitrary but fixed “sandwich” relation. R.J. Plemmons and M. West have characterized Green's relations in the usual semigroup of binary relations, and we use these to investigate Green's relations in our semigroups. We give algorithms for constructing idempotents and regular elements in these new semigroups.     

Tolerance relations on eventually regular semigroups

A semigroup is called eventually regular if a power of each element is regular. Regular and group-bound semigroups are each eventually regular. A tolerance relation on an eventually regular semigroup is introduced, and Lallement's result for regular semigroups is generalized to eventually regular semigroups. Weakly compatible tolerances on semigroups are studied. This research was partially supported by the National Science Foundation of Qufu Normal University. The author wishes to thank T. E. Hall for his comments and help in preparing this paper for publication.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

Perfect elements in Dubreil-Jacotin regular semigroups

IfS is a strong Dubreil-Jacotin regular semigroup thenx∈S is said to beperfect ifx=x(ξ∶x)x where ζ is the bimaximum element ofS. It is shown that the setP(S) of perfect elements is an ideal ofS, and is also a strong Dubreil-Jacotin subsemigroup. It is then proved that every element ofS is perfect if and only ifS is naturally ordered. Finally, necessary and sufficient conditions forP(S) to be orthodox are determined.     

Endomorphism semigroups of 2-nilpotent binary relations

We define two constructions of semigroups, which up to isomorphism describe the structure of all endomorphism semigroups of 2-nilpotent binary relations.     

Property of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions

We study properties of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions.     

Green's equivalences in finite semigroups of binary relations

This paper contains an algorithm which, given a set of generators of a semigroupS of binary relations on a finite set, computes the structure ofS in terms of Green's equivalences. The algorithm is a generalization to semigroups of binary relations of an algorithm obtained by Lallement and McFadden for semigroups of transformations. Part of this research was supported by a Mary Washington College Faculty Development Grant.     

Representations of distributive lattice-ordered semigroups with binary relations

LetR(, , ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(, , ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, , , ) where (A, , ) is a distributive lattice, (A, ) is a semigroup and is additive w.r.t. . We prove thatDLOS is the variety generated byR(, , ¦), and moreover, if (A, , , ) DLOS then it is representable whenever we disregard one of its operations.Presented by Boris M. Schein.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

Idempotents in compact monothetic semigroups

Idempotent structure of compact monothetic semitopological (separately continuous) semigroups is investigated by the methods of harmonic analysis. The pathology is shown to be arbitrarily bad in a sense made precise.     

Idempotents in compact semigroups and Ramsey theory

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson. Research supported by the National Science Foundation under Grant No. DMS86-05098.     

The Green relations approach to congruences on completely regular semigroups

On the congruence lattice (2(5) of a completely regular semigroup S the following mappings are considered _p: P and _p: VP, whereP is any of the Green relationsH, L, R orD. The equivalence relationsP andP ^V induced by these maps represent the main object of study in the paper. The former is a complete -congruence whereas the latter is a complete congruence onC(S). In particularH ,H ^V,L ^V,R ^V coincide with the kernel, trace, left trace and right trace relations onC(S), respectively. All essential properties known for the latter relations carry over to the new relationsP andP ^V. In addition, some interesting interplays of these provide for more richness in the theory of congruences on completely regular than is the case for the kernel-trace approach to congruences on regular semigroups.     

Ranks of ideals in inverse semigroups of difunctional binary relations

The set \(\mathcal {D}_n\) of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (Publ Math Debrecen 78(2):253–282, 2011), which asks: What is the rank (smallest size of a generating set) of \(\mathcal {D}_n\)? Specifically, we show that the rank of \(\mathcal {D}_n\) is \(B(n)+n\), where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of \(\mathcal {D}_n\). Although \(\mathcal {D}_n\) bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of \(\mathscr {J}\)-classes), we note that the fast growth of \({\text {rank}}(\mathcal {D}_n)\) as a function of n is a property not shared with these other families.