A Characterization Of Fuzzy Subgroups
Abbreviated Journal Title
Fuzzy Sets Syst.
Computer Science; Theory & Methods; Mathematics; Applied; Statistics; Probability
An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)⩾T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b −1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with T =‘minimum’" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">T =‘minimum’ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.
Fuzzy Sets and Systems
Anthony, J. M. and Sherwood, H., "A Characterization Of Fuzzy Subgroups" (1982). Faculty Bibliography 1980s. 177.