A left semimodule over a semiring S is an algebra [.sub.S]M = (M, +, 0, [(s*)|.sub.s[member of]S]) such that (M, +,0) is a commutative monoid and the following identities hold for all s, s' [member of] S, m, m'< [member of] M:
Let S be a semiring and [.sub.S]M a left semimodule. For A [[subset].bar] S and U [[subset].bar] M, we define
Since a semiring is a semimodule over itself, Definition 4 also defines the product of two subsets of a semiring.
Particularly, They have shown that scalar product operations can be regarded as semimodule
actions and algebraic structures like ordered idempotent semimodules
of fuzzy soft sets over ordered semirings can be constructed .
Given a Hankel matrix H(f, ) we associate with it the semimodule MH(f, ) generated by its rows.
Using this notion of linear independence, we define the notions of basis and dimension as in Guterman (2009); Cuninghame-Green and Butkovic (2004): a basis of a semimodule U over a semiring S is a set P of linearly independent elements from U which generate it, and the dimension of a semimodule U is the cardinality of its smallest basis.
Given a Hankel matrix H(f, ) with its associated semimodule MH(f, ), we define the row-rank r(H(f, )) of the matrix as the dimension of MH(f, ).
One Digital Signature Scheme in Semimodule
over Semiring // Informatica.--Lithuanian Academy of Sciences, 2005.--Vol.
From the above definition, we can easily verify that every simple semimodule over a semiring is a Euclidean semimodule.
Clearly, if A is a Euclidean semimodule over a semiring R, then any nonzero subtractive subsemimodule of A is a Euclidean R-semimodule.
Positselski, Homological algebra of semimodules
In [8, 9] such subsets were called 0-normalized semimodules
over the semigroup generated by m and n.