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

摘要

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 Cn-1, where C-n is the nth 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 d(l,m) and for a fixed endomorphism ao of a considered right ideal, the set of endomorphisms alpha such that d(l,m) (alpha) = alpha(0) is denoted by integral alpha(0)d(l,m). The last set is a semiring if and only if alpha(0) is an idempotent. The number of the semirings integral alpha(0)d(l,m), where 1 <= m <= n - 1, is equal to F-2m, which is the 2mth Fibonacci number.