We introduce “π-versions” of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π if a∧(b∨c) 6 (a∧b)∨c for all 3-element antichains {a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chain...

Equational unification is central to automated deduction and its applications in areas such as symbolic protocol analysis. In particular, the unification problem for the theory AC (“Associativity-Commutativity”) and its extensions ACI (“AC plus Idempotence”) and ACUI (“ACI with Unit element”) have been studied in great detail in the past. Distributivity (of one binary operator over another) has...

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on ∆-bonds with inva...

Recently, the distributivity of fuzzy implications over t-norms, t-conorms and uninorms was studied in many articles. In this paper we characterize functions which satisfy one of the four generalized distributivity equations of classical implication in the case when t-conorms are continuous and Archimedean. Using the obtained characterizations we describe some solutions which are fuzzy implicat...

Recently, many works have appeared dealing with the distributivity of fuzzy implications over t-norms, tconorms and uninorms (see [2, 3, 4, 5, 12, 13, 14]). These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems (see [6]). In this work we present some results connected with two functional equations describing the di...

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on ∆-bonds with inva...

The subtyping relation in the polymorphic second-order-calculus was introduced by John C. Mitchell in 1988. It is known that this relation is undecidable, but all known proofs of this fact strongly depend on the distributivity axiom. Nevertheless it has been conjectured that this axiom does not innuence the undecidability. The present paper shows undecidability of subtyping when we remove distr...

In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, di...