A question of B. Plotkin about the semigroup of endomorphisms of a free group

Abstract

Let F F be a free group of finite rank n ≥ 2 n \geq 2 , let E n d ( F ) End(F) be the semigroup of endomorphisms of F F , and let A u t ( F ) Aut(F) be the group of automorphisms of F F . Theorem. If T : E n d ( F ) → E n d ( F ) T : End(F) \to End(F) is an automorphism of E n d ( F ) End(F) , then there is an α ∈ A u t ( F ) \alpha \in Aut(F) such that T ( β ) = α ∘ β ∘ α − 1 T(\beta ) = \alpha \circ \beta \circ \alpha ^{-1} for all β ∈ E n d ( F ) \beta \in End(F) .

The automorphisms of endomorphism semigroups of free burnside groups

“…In particular, we prove, that the groups Aut(End(B(m, n))) and Aut(End(B(k, n))) are isomorphic if and only if m = k. This is a particular problem about End(A), for A a free algebra in a certain variety, was raised by B.I.Plotkin in [14]. Analogous problems for End(F ) with F a finitely generated free group or free monoid were solved by Formanek in [12] and Mashevitzky and Schein in [13] respectively.…”

Automorphisms of the Category of the Free Nilpotent Groups of the Fixed Class of Nilpotency

“…In the case when Θ is a variety of all groups we have the classical results which let us resolve this problem by an indirect way. In [3] proved that for every free group Fi the group Aut (AutFi) coincides with the group Inn (AutFi), from this result in [4] was concluded that Aut (EndFi) = Inn (EndFi) and from this fact by theorem of reduction [2] it can be concluded that AutΘ 0 = InnΘ 0 . In the case when Θ = N d is a variety of the all nilpotent class no more then d groups we know by [6], that if the number of generators i of the free nilpotent class d group N F d i are bigger enough than d, then Aut AutN F d i = Inn AutN F d i .…”

On the Monoid of Unital Endomorphisms of a Boolean Ring

“…Schreier [4] and Mal'cev [5] have described all automorphisms of End(X), where X is a set, and Gluskin [6] has described the automorphisms of End(V), where V is a vector space. More examples are provided by, among others, Formanek [7], Levi [8,9], Liber [10], Magill [11], Mashevitzky and Schein [12], Schein [13], Sullivan [14], and Sutov [15]. Recently, the subject has attracted renewed attention owing to its links with universal algebraic geometry (see, for instance, [16]).…”

“…For this reason, we focus on the power set ring. Our work is motivated on one hand by [3,[6][7][8]11,17,[19][20][21][22][23][24], where the authors have studied some questions related to the semigroup of endomorphisms of some rings, especially Boolean rings, and on the other hand by the importance of finding automorphisms of some semigroups and some (finite) rings, as explained above.…”