Congruences and Homomorphisms on Ω -algebras

Abstract

The topic of the paper are -algebras, where is a complete lattice. In this research we deal with congruences and Ω-homomorphisms. An -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce -valued congruences, corresponding quotient -algebras and -Ω-homomorphisms and we investigate connections among these notions. We prove that there is an -Ω-homomorphism from an -algebra to the corresponding quotient -algebra. The kernel of an -Ω-homomorphism is an -valued congruence. When dealing with cut structures, we prove that an -Ω-homomorphism determines classical Ω-homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an -congruence determines a closure system of classical congruences on cut subalgebras. Finally, identities are preserved under -Ω-homomorphisms.