The P-Algebras

Following the notions of Omega-set and Omega-algebra where Omega is a complete lattice, we introduce P-algebras, replacing the lattice Omega by a poset P. A P-algebra is a classical algebraic structure in which the usual equality is replaced by a P-valued equivalence relation, i.e., with the symmetric and transitive map from the underlying set into a poset P. In addition, this generalized equality is (as a map) compatible with the fundamental operations of the algebra. The diagonal restriction of this map is a P-valued support of a P-algebra. The particular subsets of this support, its cuts, are classical subalgebras, while the cuts of the P-valued equality are congruences on the corresponding cut subalgebras. We prove that the collection of the corresponding quotients of these cuts is a centralized system in the lattice of weak congruences of the basic algebra. We also describe the canonical representation of P-algebras, independent of the poset P.