Some Direct Product Of Semigroups Congruence

In this paper, we first defined a new kind of semigronp, namely left(right) strongly π-inverse semigroup, introduced and characterized the necessary and sufficient conditions for a semigroup to be left(right) strongly π-inverse, and secondly characterized the necessary and sufficient conditions for S x T to he left strongly π-inverse, determined the relation of the least group congruences between sernidirect product and the two initial semigroups, then discussed the wreath product and the standard wreath product of this new kind of semigroup. After that, we characterized the necessary and sufficient conditions for S x, T to be π-orthodox, furthermore, we characterized the necessary and sufficient conditions for S Xc. T to be right strongly π-inverse. Then we discussed the semidirect product problems of right Clifford semigroup and right Clifford π-regular semigroup. In the end, we investigated the necessary and sufficient conditions for relation on a strongly π-inverse semigroup to be a r-semiprime congruence relation, then characterized the least group congruence on a π-regular semigroup by means of the idempotents means and a relation R.