Globals of Completely Regular Monoids

An element of a semigroup S is called irreducible if it cannot be expressedas a product of two elements in S both distinct from itself. In this paper we showthat the class C of all completely regular monoids with irreducible identity elementssatises the strong isomorphism property and so it is globally determined.

One Class of the Least Congruences on Completely Regular Semigroups

Congruence is a very important aspect in the study of the semigroup theory.In general,the Kernel-trace characterizations,Green's relations and subvarieties are main tools in the consideration of congruences on completely regular semigroups.In this paper,we give one class of congruences on completely regular semigroups with the representation of wreath product of translational hulls on completely simple semigroups.By this new way,the least Clifford semigroup congruences on completely regular semigroups are generalized.

An Expository on Some Nonstandard Compactifications

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

On the Enhanced Power Graph of a Semigroup

The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.

On the Construction of Regular Orthocryptogroups

The aim of this paper is to study regular orthocryptogroups. After obtaining some charac- terizations of such semigroups, we establish the construction theorem of regular orthocryptogroups. As an application, we give the construction theorem of right quasi-normal orthocryptogroups and study homomorphisms between two regular orthocryptogroups.