semimodules

A semimodule is an algebraic structure consisting of a set S and two binary operations, usually called addition (+) and scalar multiplication (*), such that S forms a commutative monoid under addition, and scalar multiplication satisfies certain distributive and associative properties with respect to addition and an identity element. Specifically, if the scalars are from a semiring R, then (S, +) is a commutative monoid with an identity element 0. Scalar multiplication is associative: for any r, s in R and x, y in S, (r*s)*x = r*(s*x). Scalar multiplication distributes over addition in R: r*(x+y) = r*x + r*y. Also, scalar multiplication distributes over addition in S: (r+s)*x = r*x + s*x. Finally, the multiplicative identity of R acts as an identity for scalar multiplication: 1*x = x for all x in S.

Semimodules generalize the concept of modules over a ring. In a module, the scalars form a ring, which has additive inverses. In a semimodule, the scalars form a semiring, which may not have additive inverses. This relaxation allows semimodules to be used in a wider range of applications, particularly in areas where negativity is not naturally present or desirable, such as in the study of automata, formal languages, and optimization problems. Examples of semimodules include vector spaces over fields (which are also semirings), the set of natural numbers with standard addition and multiplication, and the set of tropical semirings. The study of semimodules often focuses on concepts analogous to those in module theory, such as sub-semimodules, homomorphisms, and direct sums.