subsemimodule

A subsemimodule is a substructure within a semimodule. A semimodule is an algebraic structure consisting of a set equipped with two binary operations, typically called addition and multiplication, satisfying certain axioms. These axioms often resemble those of a module or a vector space, but may be less restrictive. For instance, a semimodule might not require the existence of additive inverses for all its elements.

A subset of a semimodule is called a subsemimodule if it is closed under both addition and scalar multiplication defined by the parent semimodule. In simpler terms, if you take any two elements from the subset and add them together, the result must also be in the subset. Similarly, if you take any element from the subset and multiply it by a scalar (an element from the underlying semiring of the semimodule), the product must also remain within the subset.

The concept of a subsemimodule is analogous to that of a submodule in module theory or a subspace in vector space theory. It represents a "smaller" semimodule contained within a larger one, inheriting its algebraic properties. Studying subsemimodules can provide insights into the structure of the parent semimodule, much like studying subspaces helps understand linear transformations and vector spaces. Properties such as the intersection of subsemimodules forming a subsemimodule are often considered.