Bases in semilinear spaces over join-semirings |
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Authors: | Shan ZhaoXue-ping Wang |
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Affiliation: | College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, People's Republic of China |
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Abstract: | This paper investigates the cardinality of a basis in semilinear spaces of n-dimensional vectors over join-semirings. First, it introduces the notion of an irredundant decomposition of an element in a join-semiring, then discusses the cardinality of a basis and proves that the cardinality of each basis is n if and only if the multiplicative identity element 1 is join-irreducible. If 1 is not a join-irreducible element then each basis need not have the same number of elements in semilinear spaces of n-dimensional vectors over join-semirings. This gives an answer to an open problem raised by Di Nola et al. in their work Algebraic analysis of fuzzy systems, Fuzzy Sets and Systems 158 (2007) 1-22]. |
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Keywords: | Semiring Join-semiring Semilinear space Basis |
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