Bases in semilinear spaces over zerosumfree semirings

Standard orthogonal vectors in semilinear spaces and their applications

“…Then we can construct an L-semilinear space as follows. [15]). (a) Let L= L, +, ·, 0, 1 be a semiring.…”

“…Thus from In classical linear algebra, every set of linearly independent vectors can be orthogonalized. However, by Theorem 3.4 we have: [15]). In L-semilinear space V n , if dim(V n ) = n, a set {a 1 , a 2 , .…”

“…1 (just replace the addition and the multiplication in Definition 2.1 by the join and meet in the lattice as Fig. 1, respectively), from Theorem 3.1 of [15] we know that each basis of V 2 has the same number of elements since 1 is an additive irreducible element. However, by Definitions 2.5 and…”

“…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]). In this paper, we further investigate the standard orthogonal vectors in semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings, and discuss their characterizations, as applications, we first study the conditions that a set of vectors is a basis of a semilinear subspace which is generated by standard orthogonal vectors, and then prove that the analogue of the Kronecker-Capelli theorem is valid for systems of equations.…”

Strong linearly independent vectors in semilinear spaces and their applications

“…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]). In 2012, Shu and Wang showed that a set of linearly independent non standard orthogonal vectors can not be orthogonalized if it has at least two nonzero vectors, and proved that the analog of the Kronecker-Capelli theorem was valid for systems of equations when the column vectors of coefficient matrix was standard orthogonal(see [16]).…”