CHARACTERIZATION OF IDEAL SEMIGROUPS OF SEMILATTICES (Review Completed - Accepted)
In this paper mainly we have obtained characterization for a
semilattice to be an ideal Semigroup. In 1991 Garcia J.I. In his paper entitled “The congruence extension
property for algebraic semigroups” which is called an ideal congruence .In his
paper he defined an ideal semigroup as a semigroup in which every congruence is
an ideal congruence(Rees congruence) and has studied about the properties of
ideal semigroups.
In this paper we characterized the ideal semigroups of semilattices and cyclic
semigroups. In this paper it is observed that a semilattice E is is an ideal if and
only if every non zero element of E is maximal element. Further if S is a band or a
separative semigroup then it is a semilattice when S is an ideal semigroup. In his
paper Garcia has given an example of a cyclic semigroup which is not an ideal
semigroup.In view of this we have obtained necessary and sufficient condition for
a cyclic semigroup S to be an ideal semigroup , In fact the condition the condition
S is finite and has zero. It is also noted that every cyclic sub semigroup of a cyclic
ideal semigroup is an ideal semigroup but not for subsemigroups
Keywords: Ideal congruence, Ideal extention property, Ideal semigroup, congruence extention property