# Semiring

(redirected from Semimodule)

## 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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Given a Hankel matrix H(f, []) we associate with it the semimodule MH(f, []) generated by its rows.
A set of elements P from a semimodule U over a semiring S is linearly independent if there is no element in P that can be expressed as a linear combination of other elements in P.
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, []).
Given the Hankel matrix H(f, []) we denote by MH(f, []) the semimodule generated by the rows of H(f, []).
A Euclidean semimodule over a semiring R is a natural extension of a Euclidean semiring.
That is, a semimodule A over a semiring R is said to be unitary provided 1a = a for all a in A and the identity element 1 of R.
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.
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