Free strict n-tuple semigroups are determined by their endomorphism semigroups

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or “mirror inner". In particular, we answer a question of B. I. Plotkin.

We prove an analog of the Tits alternative for endomorphisms of \mathbb{P}^1 . In particular, we show that if S is a finitely generated semigroup of endomorphisms of \mathbb{P}^1 over \mathbb{C} , then either S has polynomially bounded growth or S contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field of any characteristic and \operatorname{Prep}(f) \not= \operatorname{Prep}(g) , then for all sufficiently large j , the semigroup \langle f^j, g^j \rangle is a free semigroup on two generators.

We prove that automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between the endomorphism semigroups of free inverse semigroups.

A doppelsemigroup is a nonempty set equipped with two binary associative operations satisfying certain identities. In this paper, we consider the variety of rectangular doppelsemigroups which are analogs of rectangular semigroups. We construct the free rectangular doppelsemigroup and characterize the least rectangular congruence on the free doppelsemigroup. As a consequence, the free rectangular semigroup is presented. We also describe all (maximal) subdoppelsemigroups, all idempotents and all endomorphisms of the free rectangular doppelsemigroup, and give a criterion for an isomorphism of endomorphism semigroups of free rectangular doppelsemigroups. In addition, we show that the endomorphism semigroup of the free rectangular doppelsemigroup is not regular in general.

Description of the space of pseudocharacters invariant with respect to a special semigroup of endomorphisms for a free semigroup of rank two