Implicational lattices and generalization of Stone's representation theorem

Let F(S) be the free algebra of type (-->, V,-->) generated by the non-empty set S, it is proved that the logical equivalent relation defined by means of R-0-semantics is a congruence relation on F(S) and the corresponding quotient algebra is said to be the R-0-semantic Lindenbaum algebra. Taking R-0-semantic Lindenbaum algebra as a prototype, the concepts of implicational lattices and regular implicational lattices which are generalizations of the concept of Boolean algebras are introduced. Besides, the concept of fuzzy implicational spaces is introduced and the representation theorem of regular implicational lattices is obtained by means of fuzzy implicational spaces. In case of Boolean algebras, the corresponding fuzzy implicational spaces are zero-dimensional compact Hausdorff spaces and here from it is proved that the famous Stone's representation theorem of Boolean algebras is a corollary of the representation theorem of regular implicational lattices.