Linear Algebra over Semirings

Many important constructions in pure and applied mathematics can be understood as semimodules over appropriate semirings. The theory of semimodules and (linear) semialgebras can be considered as a natural generalization of the theory of linear algebras and linear spaces over fields, and the theory of modules and algebras over rings. A (left) S-semimodule V over a semiring S is usually considered under assumptions that both operations of S are commutative, the addition has the (annihilating) zero 0 S , and often the multiplication has the unity element 1 S , V is a commutative additive monoid (or, more generally, only a semigroup), and, moreover, the usual conditions (see Section 3.1 below) are fulfilled. Every commutative monoid (with zero) is, of course, an ℕ-semimodule, i.e. a semimodule over the semiring ℕ of natural numbers. Any so-called in ℤ n is of necessity a left ℕ-semimodule. It is worth recalling that D. Hilbert [1890] showed that every polyhedral monoid has a finite basis (a constructive method of finding such a basis is due to A. Bachem [1978]). More general considerations concern so-called S-acts or polygons over a semiring S, in the terminology used by L.A. Skornyakov ([1973], [1978]) and investigated by him and his co-workers. See the book by M. Klip, U. Knauer & A.V. Mikhalev [2000]. The theory of matrices over semirings is intensively developed. There are also interesting investigations of the theories of semialgebras and topological semialgebras.