We prove the existence of a fixed point of non-expanding fuzzy multifunctions in -fuzzy preordered sets. Furthermore, we establish the existence of least and minimal fixed points of non-expanding fuzzy multifunctions in -fuzzy ordered sets.
Given a locale and a join semilattice with bottom element , a new concept called -slice is defined,where is as an action of the locale on the join semilattice . The -slice adopts topological properties of the locale through the action . It is shown that for each , is an interior operator on .The collection is a Priestly space and a subslice of -. If the locale is spatial we establish an isomorphism between the -slices and . We have shown that the fixed set of ,...
A lattice is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element such that at most half of the elements of satisfy . Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let denote the number of nonzero join-irreducible elements of . It is well-known that consists of at most elements....
The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations.