Partially-Ordered Semirings

Abstract

Many of the semirings originally studied, such as ℕ and ideal(R), have a partial-order structure in addition to their algebraic structure and, indeed, the most interesting theorems concerning them make use of the interplay between these two structures. In is therefore natural for us to study semirings, and semimodules over them, on which a partial order is defined. A hemiring (R, +, •) is partially-ordered if and only if there exists a partial order relation ≤ on R satisfying the following conditions for elements r, r′, and r″ of R:

• (1) If r ≤ r′ then r + r″ ≤ r′ + r″;

• (2) If r ≤ r′ and r″ ≥ 0 then rr″ ≤ r′r″ and r″r ≤ r″r′.