Projections on right and left ideals of endomorphism semiring which are derivations

Abstract

The aim of this paper is the investigation of the derivations in an endomorphism semiring of a finite chain. Such semiring can be represented as a simplex and its subsimplices are left ideals of the semiring. We construct projections on these left ideals and prove that they are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. Consequently, we obtain some results concerning nilpotent endomorphisms and using well-known result of Stanley we prove that order of semiring of nilpotent endomorphisms is equal to [Formula: see text], where [Formula: see text] is the [Formula: see text]th Catalan number. We consider a class of right ideals of the semiring and introduce projections on these ideals which are derivations and also find the maximal subsemirings of the simplex which are the domains of the constructed derivations. For one of these derivations [Formula: see text] and for a fixed endomorphism [Formula: see text] of a considered right ideal, the set of endomorphisms [Formula: see text] such that [Formula: see text] is denoted by [Formula: see text]. The last set is a semiring if and only if [Formula: see text] is an idempotent. The number of the semirings [Formula: see text], where [Formula: see text], is equal to [Formula: see text], which is the [Formula: see text]th Fibonacci number.