International Conference on Algebra in Memory of Kostia Beidar

A semiring is an algebraic structure, consisting of a nonempty set R on which we have defined two operations, addition (usually denoted by +) and multiplication (usually denoted by · or by con-catenation) such that the following conditions hold: (1) Addition is associative and commutative and has a neutral element. That is to say, a+(b+c) = (a+b)+c and a+b = b+a for all a, b, c ∈ R and there exists a special element of R, usually denoted by 0, such that a + 0 = 0 + a for all a ∈ R. It is very easy to prove that this element is unique. (2) Multiplication is associative and has a neutral element. That is to say, a(bc) = (ab)c for all a, b, c ∈ R and there exists a special element of R, usually denoted by 1, such that a1 = a = 1a for all a ∈ R. It is very easy to prove that this element too is unique. In order to avoid trivial cases, we will always assume that 1 = 0, thus insuring that R has at least two distinct elements. (3) Multiplication distributes over addition from either side. That is to say, a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c ∈ R.