Ω-algebras

In the framework of Omega-sets, where Omega is a complete lattice, we generalize the notion of a (universal) algebra, and we investigate its basic properties. Our techniques belong to the theory of lattice-valued (fuzzy) structures and we use cut-sets. An Omega-algebra is equipped with an Omega-valued equality instead of the classical one. We investigate identities and their satisfiability by these new structures. We prove that a set of identities holds on an Omega-algebra if and only if the cut-subalgebras over the corresponding cut-congruences of the Omega-valued equality satisfy the same identities in the classical setting.