Translation theory for semirings

Abstract

A semiring is a non-empty set on which two associative binary operations +, dot are defined such that dot distributes over + from the left and right. A right translation of a semiring S is an additive endomorphism rho of S such that (xy)rho = x(y rho) holds for every x, y in S. The set P(S) of all right translations of S is also a semiring under the usual addition for mappings and composition for multiplication if it is assumed that Sis subcommutative. This assumption is made throughout the paper. Chapter I develops a number of theorems about P(S), especially in relation to S. The dual development for left translations is mentioned, but not explicitly carried out. Chapter II includes several methods by which a semiring can be embedded in a semiring with identity. The translational hull for semigroups treated in Algebraic Theory of Semigroups is extended to the semiring setting. Also a theorem is given which characterizes mutually distributive semirings which are distributive lattices in terms of an embedding. In Chapter III, semirings of translations of cross product and quotient structures are discussed. A characterization of the semiring of translations of a cross product is given. The paper concludes with the demonstration of an isomorphism between the lattice of all ideals of a lattice L and the semiring of translations of L. The restriction that every sub-ideal of every principal ideal of L is a principal ideal is necessary for this theorem.