Representation of lattices by fuzzy weak congruence relations

Fuzzy (lattice valued) weak congruences of abstract algebras are investigated. For an algebra, the family of all such fuzzy relations is a complete lattice; its structure and cut properties are investigated and fully described. These fuzzy weak congruences are applied in representation of complete and algebraic lattices. A wider class of lattices can be represented in such a fuzzy framework, than in classical algebra. We prove that there is a straightforward representation of any complete lattice, using it as a co-domain. In a more general case, it is proved that several subdirect powers of lattices are also representable by fuzzy weak congruences

A note on representation of lattices by weak congruences

A weak congruence is a symmetric, transitive, and compatible relation. An element u of an algebraic lattice L is Delta-suitable if there is an isomorphism. from L to the lattice of weak congruences of an algebra such that kappa(u) is the diagonal relation. Some conditions implying the Delta-suitability of u are presented