Preliminaries

Abstract

By a semiring (S; +, •) we understand a general algebra with two binary associative operations fulfilling the following distributive laws:

$$x\cdot \left( {y + z} \right) = x\cdot y + x\cdot z$$

(1)

and

$$\left( {y + z} \right)\cdot x = y\cdot z + z\cdot x$$

(2)

for all x, y, z ∈ S. If the addition is commutative and has a neutral element 0 (i.e. if (S; +) is a commutative monoid), which is an annihilating (or absorbing) element, that

$$x + y = y + x$$

(3)

$$x + 0 = x = 0 + x$$

(4)

$$x\cdot 0 = 0 = 0\cdot x$$

(5)

for arbitrary x, y ∈ S, then (S; +, •) is said to be a hemiring (see the monographs by J.S. Golan [1992], [and such papers as K. Iizuka [1959], D.R. LaTorre [1965], D.M. Olson [1978], D.M. Olson & T.L. Jenkins [1983] and S.M. Yusuf & M. Shabir [1988]; note that there are some differences in terminology). If a semiring S has an element 0 with property (4), then S is called a semiring with zero. A semiring with zero and commutative addition is called (by some theoretical physicists and categorists) a rig (in the sense of “rings without negation”). Observe that, in general, the zero element does not need to be annihilating (see, e.g., K. Głazek [1968a]).