Linear Algebra over Semirings

Kazimierz Głazek
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Kazimierz Głazek: University of Zielona Góra, Institute of Mathematics

Chapter 3 in A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences, 2002, pp 25-42 from Springer

Abstract: Abstract 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.

Keywords: Linear Algebra; Path Algebra; Discrete Event System; Commutative Monoid; Iteration Theory (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-94-015-9964-1_4

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