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 1s 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