The Closed Subsemigroups of a Clifford Semigroup

Abstract

In this paper we study the closed subsemigroups of a Clifford semigroup. It is shown that $\{\underset{\alpha \in \overline{Y'}}{\cup} G_{\alpha} | Y' \in P(Y)\}$ is the set of all closed subsemigroups of a Clifford semigroup $S = [Y ; G_α; \phi_{α, β}]$, where $\overline{Y'}$ denotes the subsemilattice of $Y$ generated by $Y'$. In particular, $G$ is the only closed subsemigroup of itself for a group $G$ and each one of subsemilattices of a semilattice is closed. Also, it is shown that the semiring $\overline{P}(S)$ is isomorphic to the semiring $\overline{P}(Y)$ for a Clifford semigroup $S = [Y ; G_α; \phi_{α, β}]$.