A characterization of the group congruences on a semigroup

In this paper we describe the group congruences on a semigroupS in terms of their kernels. In particular, we show that the least group congruence σ on a dense and uniteryE-semigroupS is defined by, for alla, b ∈ S, (a, b) ∈ σ iff (εe, f εE(S)) ea=bf Communicated by John M. Howie     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

Subdirectly irreducible \mathcal{RGC} -commutative right H-semigroups

A semigroup S is said to be ℛ-commutative if, for all elements a,b∈S, there is an element x∈S ¹ such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba ²=a ² ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.     

Congruence triples on regular semigroups and their generalizations

In this paper, we introduce the concept of VT-congruence triples on a regular semigroup S and show how such triples can be constructed by using the equivalences on S/ℒ, S/R and the special congruences on S. Also, such congruence triples are characterized so that an associated congruence can be uniquely determined by a given congruence triple. Moreover, we also consider the VH-congruence pairs on an orthocryptogroup.     

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.     

Representation of congruences on regular semigroups with inverse transversals

For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S ^o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component partsI,S ^o and Λ. The structure of images of this type of semigroups is also presented. This work is supported by Natural Science Foundation of Guangdong Province     

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

On semirings whose additive reduct is a semilattice

A semiring S whose additive reduct is a semilattice is called a k-regular semiring if for every a∈S there is x∈S such that a+axa=axa. For a semigroup F, the power semiring P(F) is a k-regular semiring if and only if F is a regular semigroup. An element e∈S is a k-idempotent if e+e ²=e ². Basic properties of k-regular semirings whose k-idempotents are commutative have been studied.     

S-Classification of Regular Semigroups

On any regular semigroup S, the greatest idempotent pure congruence τ the greatest idempotent separating congruence μ and the least band congruence β are used to give the S-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category S whose morphisms are surjective K- and T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category S whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from S to S. The effect of the S-classification on Reilly semigroups and cryptogroups is discussed briefly.     

Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms are permutable.     

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.     

On Isomorphisms of Minimal Cayley Graphs and Digraphs

 A Cayley graph or digraph Cay(G,S) is called a CI-graph of G if, for any T⊂G, Cay(G,S)≅Cay(G,T) if and only if S ^σ=T for some σ∈Aut(G). The aim of this paper is to characterize finite abelian groups for which all minimal Cayley graphs and digraphs are CI-graphs. Received: February 13, 1998 Final version received: May 7, 1999     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

Regular semigroups with skew pairs of idempotents

Let S be a regular semigroup with set of idempotents E(S) . Given x,y ∈ S , we say that (x,y) is a skew pair if x y \notin E(S) whereas y x ∈ E(S) . Here we use this concept to characterise certain regular Rees matrix semigroups.     

On the number of vertices and edges of the Buneman graph

In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.