Equivalential algebras with conjunction on the regular elements

We introduce the definition of the three-element equivalential algebra R with conjunction on the regular elements. We study the variety generated by R and prove the Representation Theorem. Then, we construct the finitely generated free algebras and compute the free spectra in this variety.

FREGEAN VARIETIES

A class [Formula: see text] of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in [Formula: see text] are uniquely determined by their 0–cosets and ΘA (0,a) = ΘA (0,b) implies a = b for all [Formula: see text]. The structure of Fregean varieties is investigated. In particular it is shown that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e. algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. Moreover the clone of polynomials of any finite algebra A from a congruence permutable Fregean variety is uniquely determined by the congruence lattice of A together with the commutator of congruences. Actually we show that such an algebra A itself can be recovered (up to polynomial equivalence) from its congruence lattice expanded by the commutator, i.e. the structure ( Con (A); ∧, ∨, [·,·]). This leads to Fregean frames, a notion that generalizes Kripke frames for intuitionistic propositional logic.

Free spectra of linear equivalential algebras

We construct the finitely generated free algebras and determine the free spectra of varieties of linear equivalential algebras and linear equivalential algebras of finite height corresponding, respectively, to the equivalential fragments of intermediate Gödel-Dummett logic and intermediate finite-valued logics of Gödel. Thus we compute the number of purely equivalential propositional formulas in these logics in n variables for an arbitrary n ∈ ℕ.