W. B. Vasantha Kandasamy, Florentin Smarandache
In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup. The semigroup under × has no finite subsemigroups but under min or max we have subsemigroups of order one, two and so on. [0, 1) under + modulo 1 is a group and [0, 1) has finite subgroups. We study [0, 1) with two binary operations min and max resulting in semiring of infinite order. This has no subsemirings which is both an ideal and a filter. However pseudo semiring under min and × has subsemirings which is both a filter and an ideal. Construction and study of pseudo rings on [0, 1) is interesting as distributive law is not true. Study of algebraic structures on the fuzzy interval [0, 1) is innovative and interesting.
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