Fuzzy Ideal and Fuzzy Filter on Ordered Groupoid
DOI:
https://doi.org/10.20956/912h0e65Keywords:
Ordered Groupoid, Fuzzy Ideal, Fuzzy Filter, Fuzzy Membership Function, Fuzzy Prime IdealAbstract
Within fuzzy set theory, every element of nonempty set is assigned membership degree in interval [0, 1]. With development of fuzzy set theory, its concepts have been extended to algebraic structures, including ordered groupoids, leading to investigation of fuzzy ideals and fuzzy filters. This research investigates properties of fuzzy ideals together with their correspondence to classical ideals through fuzzy membership functions. It also studies the characteristics of fuzzy filters and their connection with fuzzy prime ideals. This research uses literature review to examine relationship between classical ideals and fuzzy ideals in ordered groupoids and by proving conditions for fuzzy subset to be a fuzzy ideal. Analysis also includes the characterization of fuzzy filters and their relation to fuzzy prime ideals through concept of fuzzy subset complements. The findings reveal that the characteristic function induced by a nonempty subset forms a fuzzy ideal precisely when the subset itself is an ideal of the ordered groupoid. Therefore, fuzzy ideals provide a natural generalization of classical ideals while maintaining the ordering relation and binary operation defined on an ordered groupoid.
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