Abstract: We give a brief survey of the representation of an inverse semigroup, especially of the representation of a Clifford semigroup with identity (i.e., the Clifford monoid). First we give a short background of the representation of finite groups, the inverse semigroups and the Clifford monoid which is a regular monoid as a semilattice of groups. Hopf algebras and Semilattice graded weak Hopf algebras can be considered as the generalization of group algebras and Clifford monoid algebras, respectively. We describe how the semilattice graded weak Hopf algebra can be considered as the generalization of Hopf algebra as the Clifford monoids are considered as the generalizations to groups. In this note, we also discuss the Clifford monoid algebra and its relationship with the semilattice graded weak Hopf algebra like the group algebra has a relationship with the Hopf algebra. We shed light on the importance of the representation of Clifford monoids, its algebras and the representation of Semilattice graded weak Hopf algebras. One of the main objects of this note is to get inspired from the rich structure theory of groups to make it possible develope the theory of inverse semigroups and of Clifford semigroups with identity. The second object is to go a step further in jumping from the group of grouplike elements of a Hopf algebra and getting motivation to obtain the Clifford semigroup with identity which is the set of grouplike elements of a semilattice graded weak Hopf algebra. The main purpose of this survey is to get inspiration to develope the representation theory of weak Hopf algebras and of semilattice graded weak Hopf algebras and characterizing such algebras. |