Abstract
Publisher Summary In the past decades, matrices with entries from various semirings have attracted attention of many researchers working in both theoretical and applied mathematics. A semiring is a set S with two binary operations, addition and multiplication, such that S is an Abelian monoid under addition (identity denoted by 0), S is a semigroup under multiplication (identity, if any, denoted by 1), multiplication is distributive over addition on both sides, and s 0 = 0 s = 0 for all s ∈ S . It is assumed that there is a multiplicative identity 1 in S , which is different from 0. The development of linear algebra over semirings certainly requires an analog of the determinant function. However, it turns out that even over commutative semirings without zero divisors, the classical determinant cannot be defined as over fields and commutative rings. The main problem lies in the fact that in semirings that are not rings not all elements possess an additive inverse.
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