Fuzzy group theory: A comparison of different notions of product of fuzzy sets

The notion of penultimate subgroup of a fuzzy group is introduced. This leads to the concept of penultimate product of fuzzy subgroups. A detailed study of different notions of product of fuzzy sets in a groupoid is carried out. The ultimate superiority of Liu's set product is established. Level subsets and strong level subsets of these products are also analysed. Consequently, simplified proofs of some of the results on free product are obtained.