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Our work extensively depends upon the regular monoids, the Clifford monoids, their algebras, the Hopf algebras and the weak Hopf algebras(Lis ). By M. Petrich [74], the class of all Clifford monoids form a variety of inverse monoid. Further, we can say that the class of idempotents of all Clifford monoids form the subvariety of all Clifford monoids. A. H. Clifford in [25] proved the famous structure theorems; Theorem 4.1.3 and Theorem 4.2.1 for the semigroups which Clifford used the name semigroups admitting relative inverses in [25], by virtue of these results Howie first time named such semigroups, the Clifford semigroups in [42]. It has been structured by Clifford in [25] and latter Howie described in [42] that a Clifford semigroup is a strong semilattice of groups. By [74] and [42], if S is a Clifford monoid then S is disjoint union of its maximal subgroups.
F. Li introduced the so-called weak Hopf algebra in 1999 (see [55]), latter he defined semilattice graded weak Hopf algebra. He laid its foundation in his latter work, (see [55], [56], [57], [58], [59], [60], [61] and [62]). Moreover, it is known that the regular monoid algebra and the Clifford monoid algebra are the respective examples of the weak Hopf algebra and the semilattice graded weak Hopf algebra. Weak Hopf algebra can be considered as the generalization of Hopf algebra. The set of grouplike elements of the Hopf algebra is group, however the grouplike elements of the weak Hopf algebra is a regular monoid and that of semilattice graded weak Hopf algebra is a Clifford monoid. The Clifford monoid is an inverse semigroup(with 1) whose idempotents lies in its center. Thus a Clifford monoid may also be named as generalized group in the Vagners sense [89] and [90].
We discuss Clifford monoid algebra which is a natural example of the semi- lattice graded weak Hopf algebra. We shed light on the structures of Clifford monoids, its algebras and the structures of Semilattice graded weak Hopf alge- bras using quivers. One of the main objects of this work is to make it possible develop the theory of weak Hopf algebras. Another purpose of the work is to de- velop the structure theory of weak Hopf algebras and of semilattice graded weak Hopf algebras using path coalgebras corresponding to some quivers. One may use the structure theory and the characterization theory of Clifford monoids algebras and of Hopf algebras to achieve this task.