Clifford semigroup actions

In this paper, we establish the orbit-stabilizer theorem and the orbit decomposition theorem for a Clifford semigroup. Moreover, we establish that for a Clifford semigroup [Formula: see text], any two homogeneous Clifford [Formula: see text]-sets [Formula: see text] and [Formula: see text] are Clifford [Formula: see text]-isomorphic if and only if for any [Formula: see text], [Formula: see text] is conjugate to [Formula: see text]. More generally, for a Clifford semigroup [Formula: see text], any two Clifford [Formula: see text]-sets [Formula: see text] and [Formula: see text] are Clifford [Formula: see text]-isomorphic if and only if there is a bijection [Formula: see text] satisfying the property that, for all [Formula: see text], [Formula: see text] is Clifford [Formula: see text]-isomorphic to [Formula: see text]. Finally, we prove that in a Clifford semigroup [Formula: see text], for any element [Formula: see text] such that [Formula: see text] is [Formula: see text]-unitary full Clifford subsemigroup of [Formula: see text], [Formula: see text], where [Formula: see text] is the conjugacy class in [Formula: see text] containing the element [Formula: see text].