Semiring

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Sem´i`ring`

    (sĕm´ĭ`rĭng`)
n.1.(Anat.) One of the incomplete rings of the upper part of the bronchial tubes of most birds. The semirings form an essential part of the syrinx, or musical organ, of singing birds.
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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 [26].
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, []).
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 and semicontramodules.
In [8, 9] such subsets were called 0-normalized semimodules over the semigroup generated by m and n.