Abstract
An associative product, to be called the free product, of fuzzy sets in a semigroup is defined, which lies between sup-min and sup-max products. It behaves nicely for characteristic functions and fuzzy point sets and simulates all the standard results for the product of two subgroups of a group. Instead of fuzzy invariance, which is too strong, only a milder form of relative fuzzy invariance is used to establish fuzzy subgroupness of the product, which is intimately connected with the necessity of two fuzzy subgroups to commute. Lastly, under a slightly restricted situation, the free product induces a fuzzy subgroup on a quotient group. It is suggested that the product be tested for a suitable definition of a fuzzy topological group. An analysis of its deeper aspects is forthwith.
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