Representations through a monoid on the set of fuzzy implications

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set II of all fuzzy implications. To this end, we propose a binary operation, denoted by ⊛, which makes (I,⊛I,⊛) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup SS of this monoid and using its representation define a group action of SS that partitions II into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representations of the two main families of fuzzy implications, viz., the f- and g-implications.