Standard Bases of a Vector Space Over a Linearly Ordered Incline |
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Authors: | Jun-Sheng Duan Ai-Ping Guo Fen-Xia Zhao Li Xu Wen-Guang Tang |
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Affiliation: | 1. College of Science, Shanghai Institute of Technology , Shanghai, China duanjssdu@sina.com;3. School of Mathematics, Baotou Teachers College , Baotou, China;4. College of Science, Tianjin University of Commerce , Tianjin, China |
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Abstract: | Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described. |
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Keywords: | Basis Incline Semiring Standard basis Vector space |
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