Invertible matrices and semilinear spaces over commutative semirings

Standard orthogonal vectors in semilinear spaces and their applications

“…For all a, b ∈ L, if r = a + b implies that r = a or r = b, then r is called an additive irreducible element of semiring L (see [17]). The following definition is taken from [15].…”

“…In 2007, Di Nola et al used the notions of semirings and semimodule to introduce the concept of semilinear space in the MV-algebraic setting, and obtained some similar results as those of classical linear algebras (see [7]). In 2010, Perfilieva and Kupka showed that the necessary condition of the Kronecker-Capelli theorem is valid for systems of equations in a semilinear space of n-dimensional vectors (see [12]), Zhao and Wang gave a sufficient condition that each basis in semilinear spaces of n-dimensional vectors has the same number of elements over commutative zerosumfree semirings (see [17]), moreover, in 2011, they obtained a necessary and sufficient condition that each basis has the same number of elements over join-semirings (see [18]), where a join-semiring is just a kind of zerosumfree semiring. In 2011, Shu and Wang showed some necessary and sufficient conditions that each basis has the same number of elements over commutative zerosumfree semirings and proved that a set of vectors is a basis if and only if they are standard orthogonal (see [15]).…”

“…Example 2.3 (Zhao and Wang[17]). The following are examples of commutative zerosumfree semirings:(i) The real interval [0, 1] under the operations a + b = max{a, b} and a · b = min{a, b} for all a, b ∈ [0, 1];…”

Strong linearly independent vectors in semilinear spaces and their applications

“…In 2007, Di Nola et al used the notions of semirings and semimodule to introduce the concept of semilinear space in the MV-algebraic setting, and obtained some similar results as those of classical linear algebras (see [7]). In 2010, Perfilieva and Kupka showed that the necessary condition of the Kronecker-Capelli theorem is valid for systems of equations in a semilinear space of n-dimensional vectors (see [12]), Zhao and Wang gave a sufficient condition that each basis in similinear spaces of ndimensional vectors has the same number of elements over commutative zerosumfree semirings (see [20]), moreover, 1n 2011, they obtained a necessary and sufficient condition that each basis has the same number of elements over join-semirings (see [21]), where a join-semiring is just a kind of zerosumfree semiring. In 2011, Shu and Wang showed some necessary and sufficient conditions that each basis has the same number of elements over commutative zerosumfree semirings and proved that a set of vectors is a basis if and only if they are standard orthogonal (see [15]).…”

A Heuristic Method to Compute the Approximate Postinverses of a Fuzzy Matrix

“…The invertibility of matrices on general algebraic structures were also discussed in [7], [9], [19], [21], and [24].…”