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The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's relations are described. Criteria of primary conjugacy and conjugacy are given for certain naturally defined families of partial automaton permutations. Sufficient conditions under which an inverse semigroup admits a level transitive action are presented. We give explicit examples (monogenic inverse semigroups and some commutative Clifford semigroups) of inverse semigroups generated by finite automata.

Let M be a Clifford monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck–Reilly extensions of Clifford monoids to be finitely presented.

The set of all endomorphisms of an algebraic structure with composition of functions as operation is a rich source of semigroups, which has only rarely been dipped into (see [2]). Here we make a start by considering endomorphisms of Clifford semigroups, relating them to the homomorphisms and endomorphisms of the underlying groups.

In this paper, we prove that the dominion of any full orthodox subsemigroup of a medial orthodox semigroup is described by the Isbell zigzag theorem in the category of medial orthodox semigroups. As a consequence, the dominions of any full completely regular subsemigroup of a medial completely regular semigroup as well as that of any full Clifford subsemigroup of a medial Clifford semigroup are also described by the Isbell zigzag theorem in the categories of all medial completely regular semigroups and of all medial Clifford semigroups, respectively.

Each completely regular semigroup is a semilattice of completely simple semigroups. The more specific concept of a strong semilattice provides the concrete product between two arbitrary elements. We characterize strong semilattices of rectangular groups by so-called disjunctions of identities. Disjunctions of identities generalize the classical concept of an identity and of a variety, respectively. The rectangular groups will be on the one hand left zero semigroups and right zero semigroups and on the other hand groups of exponent $p\in P$, where $P$ is any set of pairwise coprime natural numbers.