Importation lattices

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P≠∅, we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P=[0,1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order-theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0,1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations.