Deductive systems and R-congruences of pre-Hilbert algebras

Pre-Hilbert algebras belong to a wide class of algebras of logic, they are a generalization of well-known Hilbert algebras. Deductive systems of algebras of logic are an important algebraic notion. In the paper, the properties and characterizations of deductive systems of pre-Hilbert algebras are investigated. It is proven that the deductive systems of a pre-Hilbert algebra (with respect to ⊆) form a relatively pseudocomplemented algebraic lattice. Moreover, R-congruences are introduced and studied. It is shown that the lattice of deductive systems of a pre-Hilbert algebra A is isomorphic to the lattice of R-congruences on A. The construction of the quotient algebra A/D of a pre-Hilbert algebra A via a deductive system D of A is given and the fundamental homomorphism theorem is obtained. Furthermore, maximal deductive systems are investigated. The homomorphic properties of maximal deductive systems of a pre-Hilbert algebra are provided. Finally, some characterizations of maximal deductive systems are given.