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In his paper “On quasi-separative ‘semigroup’s’”, Krasilnikova, Yu. I. and Novikov, B. V. have studied congruences induced by certain relations on a “semigroup”. They further showed that if the “semigroup” is quasi separative then the induced congruence is a semilattice congruence. In this paper we continue the study of these relations and the induced congruences i.e., the congruences induced by certain relations on ‘‘semigroup’s”. In this paper mainly it is observed that if S is a quasi-separative and regular “semigroup” then the necessary and sufficient condition for to be the smallest semilattice congruence η is obtained.

In this paper “On quasi-separative ‘semigroup’s’”, Krasilnikova Yu. I. and Novikov B.V. have studied congruences induced by certain relations on a “semigroup”. They further showed that if the “semigroup” is quasi-se- parative then the induced congruence is a semilattice congruence. In this paper we continue the study of these relations and the induced congruences. In theorem 2, we have proved that the family of all relatios

The following definition is due to Krasilnikova Yu. I. and Novikov B. V. (see [

Def 1: Let S be a “semigroup” and Ω be a relation on S satisfying conditions.

where

Define a relation

Lemma 2: Let

Proof: Let

Lemma 3: Let

Proof: Since

Theorem 4: Let S be quasi-separative and regular “semigroup”. Then

Proof: Suppose S is quasi-separative and regular and Suppose

Corollary 5: If S is a commutative regular “semigroup” then

Corollary 6: If S is a completely regular and

The following is an example of a completely regular “semigroup” in which

Example 7: Let S be a left zero “semigroup” with at least two elements. If

Theorem 8: In a band S,

It is natural to ask whether every semilattice congruence on “semigroup” is of the form

The following example shows that it is not true.

Example 9: Consider the non modular lattice

The following example shows that

Example 10: Let

The following example shows that in non quasi-separative “semigroup’s” there exists

Example 11: Let S be a non quasi-separative “semigroup”, then 1_{s} is in

It is interesting to note that if S is a left or right zero “semigroup” then

In paper [

Example 12: Consider the “semigroup”

Then h-classes are {a, b} and {c, d} which are right zero “semigroup’s” and hence S is a semilattice of weakly cancellative “semigroup’s”, but S is not weakly balanced since

The following is an example of quasi separative “semigroup”, which is not completely regular.

Example 13: Consider the “semigroup”

where

Thorem 14: Let S be a separative “semigroup”, and

Proof: Let S be a separative “semigroup” and a Î S such that E(a) is a semilattice congruence. Then for any

We are very much thankful to the referees for their valuable suggestions.

K. V. R.Srinivas, (2015) On Congruences Induced by Certain Relations on “Semigroups”. Advances in Pure Mathematics,05,579-582. doi: 10.4236/apm.2015.59054