Free rectangular doppelsemigroups

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.

The automorphisms of endomorphism semigroups of relatively free groups

The question of describing the automorphisms of [Formula: see text] for a free algebra [Formula: see text] in a certain variety was considered by different authors since 2002. In this paper, we consider this question for the class of all relatively free groups having only cyclic centralizers of non-trivial elements. We prove that each automorphism of the endomorphism semigroup [Formula: see text] of groups [Formula: see text] from this class is uniquely determined by its action on the subgroup of inner automorphisms [Formula: see text]. The obtained general result includes the following cases: absolutely free groups, free Burnside groups of odd period [Formula: see text], free groups of infinitely based varieties of Adian (the cardinality of the set of such varieties is continuum), and so on.

A note on generators of the endomorphism semigroup of an infinite countable chain

In this note, we consider the semigroup [Formula: see text] of all order endomorphisms of an infinite chain [Formula: see text] and the subset [Formula: see text] of [Formula: see text] of all transformations [Formula: see text] such that [Formula: see text]. For an infinite countable chain [Formula: see text], we give a necessary and sufficient condition on [Formula: see text] for [Formula: see text] to hold. We also present a sufficient condition on [Formula: see text] for [Formula: see text] to hold, for an arbitrary infinite chain [Formula: see text].

Structure of Endomorphism Semigroup of Endomappings of Some Particular Rooted Trees

For an endomapping of a finite set of points lying on some rooted particular trees, endomorphism and endomorphism semigroup were studied. The main aim of this paper was to obtain the structure of the semigroup of all endomorphisms of the endomappings represented by directed graphs on particular types of rooted trees.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 2, 169-175, 2015

Automorphisms of the endomorphism semigroup of a free algebra

We suggest a new method for describing automorphisms of the endomorphism semigroup of a free algebra in a variety of algebras. As an application, we describe automorphisms of the endomorphism semigroup of a free Poisson algebra over an arbitrary field [Formula: see text].

The endomorphism semigroup of an endomapping of a finite set

The structure of the semigroup of all endomorphisms of an endomapping of a finite set has been determined. This has been done by naturally representing the endomapping by a directed graph, and determining the structure of the endomorphism semigroup of this graph.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10310GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 71-77

REPRESENTATION THEORY OF THE STABILIZER SUBGROUP OF THE POINT AT INFINITY IN Diff(S1)

The group Diff (S1) of the orientation-preserving diffeomorphisms of the circle S1 plays an important role in conformal field theory. We consider a subgroup B0 of Diff (S1) whose elements stabilize "the point at infinity." This subgroup is of interest for the actual physical theory that lives on the punctured circle, or the real line. We investigate the unique central extension [Formula: see text] of the Lie algebra of that group. We determine the first and second cohomologies, its ideal structure and the automorphism group. We define a generalization of Verma modules and determine when these representations are irreducible. Its endomorphism semigroup is investigated and some unitary representations of the group which do not extend to Diff (S1) are constructed.