具有Clifford断面的正则纯正半群

给出了具有Clifford断面的右正规纯正半群的等价刻画,得到了具有Clifford断面的正则纯正半群的次直积分解,证明了具有Clifford断面的正则纯正半群一定是正则纯正群．     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Orthodox semigroups with complemented congruence lattices

Communicated by J. M. Howie     

A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S ^* of S and a unary operation * in S satisfying: (1) x ^* ∈ S ^* \cap V_ S ^* (x) for all x∈ S; (2) (x ^* ) ^* =x for all x∈ S ^* ; (3) (x ^* y) ^* =y ^* x ^** and (xy ^* ) ^* =y ^** x ^* for all x,y∈ S, then S ^* is called a regular *- transversal of S ; if (3) is replaced with (xy) ^* =y ^* x ^* for all x,y∈ S, then S ^* is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.     

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     

Two congruence lattices of completely simple semigroups

For a Rees matrix semigroupS with normalized sandwich matrix and C(S), the congruence lattice ofS, we consider the lattice generated by {itpT_l, pK, pT_r, pt_l, pk, pt_r}. HerepT ₁ andpt _l are the upper and lower ends of the interval which makes up the _i -class of , _i being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {T _l, K, T_r, t_l, _k, t_r} where and are the equality and the universal relations onS, respectively. In both cases, we find lattices freest relative to these lattices and represent them as distributive lattices with generators and relations.With 3 Figures