Representations through a monoid on the set of fuzzy implications

The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

“…In this work, we continue our study of the recently proposed generative method [50,53], viz., the -composition (Definition 1.2) on I. While some algebraic aspects have already been explored in [53], this work can be broadly seen to discuss the closures of some subsets of I w.r.t.…”

“…In fact, the operation not only leads to new fuzzy implications but also to a rich algebraic structure, namely a non-idempotent monoid, on the set I of fuzzy implications. The algebraic aspects of the monoid (I, ) have been already explored in [53] leading upto hitherto unknown representations of the Yager's families of fuzzy implications.…”

“…While some algebraic aspects have already been explored in [53], this work can be broadly seen to discuss the closures of some subsets of I w.r.t. , or alternately, of investigating the analytical aspects of the -composition.…”

“…the -composition. Table 3, [53]) shows some new fuzzy implications obtained from some of the basic fuzzy implications listed in Table 1 via the operation defined in Definition 1.2.…”

“…Table 6 for its explicit formulae or Example 2.9 (i) in [53]). (ii) However, note that in the above example, J I satisfies (OP), since J I = J = I LK , satisfies (OP).…”