Simplices in the Endomorphism Semiring of a Finite Chain

We establish new results concerning endomorphisms of a finite chain, if the cardinality of the image of such endomorphism is no more than some fixed number. The semiring of all such endomorphisms can be seen as a simplex whose vertices are the constant endomorphisms. We explore the properties of these simplices. 1. Introduction and Preliminaries It is well known that each simplicial complex has a geometric (continuous) interpretation as a convex set spanned by geometrically independent points in some Euclidean space. Here, we present an algebraic (discrete) interpretation of simplicial complex as a subsemiring, containing (in some sense spanned by) constant endomorphisms of the endomorphism semiring of a finite chain. The endomorphism semiring of a finite semilattice is well studied in [1–10]. The paper is organized as follows. After the introduction and preliminaries, in Section 2, we give basic definitions and obtain some elementary properties of simplices. Although we do not speak about any distance here, we define discrete neighborhoods with respect to any vertex of the simplex. In Section 3, we study discrete neighborhoods, left ideals, and right ideals of a simplex. The main results are Theorems 9 and 12, where we find two right ideals of simplex. In Section 4, Theorem 15 is the main result of the paper, where we show that important objects (idempotents, -nilpotent elements, left ideals, and right ideals) of simplex (big semiring) can be constructed using similar objects of coordinate simplex (little semiring). Since the terminology for semirings is not completely standardized, we say what our conventions are. An algebra , with two binary operations + and on , is called a semiring if:(i) is a commutative semigroup;(ii) is a semigroup;(iii)both distributive laws hold and , for any . Let be a semiring. If a neutral element of the semigroup exists and or , it is called a left or a right zero, respectively, for all . If , for all , then it is called zero. An element of a semigroup is called a left (right) identity provided that or , respectively, for all . If a neutral element of the semigroup exists, it is called identity. A nonempty subset of is called an ideal if , , and . The facts concerning semirings can be found in [1]. For a join-semilattice , set of the endomorphisms of to be a semiring with respect to the addition and multiplication defined as follows:(i) , when , for all ;(ii) , when , for all .This semiring is called the endomorphism semiring of . In this paper, all semilattices are finite chains. Following [2], we fix a finite chain and