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 I of all fuzzy implications. To this end, we propose a binary operation, denoted by ~, which makes (I,~) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup S of this monoid and using their representation define a group action of S that partitions I into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representation of the two main families of fuzzy implications, viz., the f and g-implications.