Semilinear Spaces Over Semirings And The Solution Of Fuzzy Relational Equations

This paper introduces a semilinear space over semirings, defines the notions of vector, linear independence and basis, discusses matrices and bases in semilinear spaces of n-dimensional vectors over semirings, and de-scribes the solution sets of a special case of a system of fuzzy relational equa-tions. First, some necessary and sufficient conditions for a matrix to be in-vertible are obtained, it is shown that a matrix is invertible if and only if the system of its column vectors is a basis in semilinear spaces of n-dimensional vectors over commutative semirings, and the factor rank of matrix is also investigated. Then the range of the cardinality of a basis in semilinear spaces of n-dimensional vectors over semirings is discussed. It is identified that the cardinality of each basis is no less than n, and the cardinality of each ba-sis is n under some conditions. In particular, the range of the cardinality of a basis in semilinear spaces of n-dimensional vectors over MV-algebras is deeply investigated. The notion of an irredundant finite decomposition of an element in a semiring is introduced, then the precise range of the cardinality of a basis in semilinear spaces of n-dimensional vectors over MV-algebras is obtained. In the end, the solution set of a special case of a system of fuzzy relational equations in Brouwerian lattices is formulated in a similar way as that of linear spaces.