Narrow Results

By the operator of relative complementation is meant a mapping assigning to every element $x$ of an interval $[a,b]$ of a lattice $L$ the set $x^{ab}$ of all relative complements of $x$ in $[a,b]$. Of course, if $L$ is relatively complemented then $x^{ab}$ is nonempty for each interval $[a,b]$ and every element $x$ belonging to it. We study the question under what condition a complement of $x$ in $L$ induces a relative complement of $x$ in $[a,b]$. It is well-known that this is the case provided $L$ is modular and complemented. However, we present a more general result. Further, we investigate properties of the operator of relative complementation, in particular in the case when the interval $[a,b]$ is a modular sublattice of $L$ or if it is finite. Moreover, we characterize when the operator ${}^{ab}$ of relative complementation satisfies the identity ${(x^{ab})}^{ab}\approx x$ provided $[a,b]$ is complemented and we show a class of lattices where this identity holds. Finally, we establish sufficient conditions under which two different complements of a given element $x$ of $[a,b]$ induce the same relative complement of $x$ in this interval.

We characterize logical connectives given by t-norms and t-conorms which are N-complementary with respect to a strong negation. We clarify the relation between this notion and the usual N-duality as well as its implications concerning the validity of the classical-like Excluded-Middle and Non-Contradiction laws in Fuzzy Logic.