The main aim of this article is to study admissible congruences on a type B semigroup. First, we give characterizations of the minimum admissible congruence whose trace is a normal congruence on a type B semigroup. After obtaining some properties of admissible congruences with the same trace on a type B semigroup, we introduce the notion of a normal subsemigroup and give characterizations of the minimum and maximum admissible congruences whose kernels are normal subsemigroups. Finally, the concept of a congruence pair of a type B semigroup is given, and two congruences associated with the congruence pair are obtained.
The definition of an almost left factorizable weakly type B semigroup is introduced by using the notion of a left factorizable weakly type B monoids, and some properties and characterizations on such semigroup are obtained. The main aim of this paper is to give characterizations of sufficient and necessary conditions for a weakly type B semigroup to be almost left factorizable. In particular, some properties of proper almost factorizable weakly type B semigroups are given.
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