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.