By Josep M. Font, Ramon Jansana

The aim of this monograph is to enhance a really basic method of the algebra­ ization of sententiallogics, to teach its effects on a couple of specific logics, and to narrate it to different present methods, particularly to these according to logical matrices and the equational outcome constructed through Blok, Czelakowski, Pigozzi and others. the most virtue of our approachlies within the mathematical items used as versions of a sententiallogic: We use summary logics, whereas the dassical ways use logical matrices. utilizing types with extra constitution permits us to mirror in them the metalogical houses of the sentential good judgment. due to the fact an summary common sense will be seen as a "bundle" or relatives of matrices, one may perhaps imagine that the recent types are primarily reminiscent of the previous ones; yet we think, after an total appreciation of the paintings performed during this region, that it really is exactly the remedy of an summary common sense as a unmarried item that provides upward thrust to an invaluable -and attractive- mathematical thought, capable of clarify the connections, not just on the logical Ievel yet on the metalogical Ievel, among a sentential good judgment and the actual dass of types we go together with it, particularly the dass of its complete versions. routinely logical matrices were considered as the main compatible concept of version within the algebraic stories of sentential logics; and certainly this concept offers sev­ eral completeness theorems and has generated an attractive mathematical thought.

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KS is the variety generated by the class Alg∗S. ∼ P ROOF. 6), iff for any γ(p, q) ∈ F m , γ(ϕ, q ) S γ(ψ, q ). 1) amounts to saying that for all a, ϕA (a) , ψ A (a) ∈ ΩA (F ), and this is equivalent to ϕA (a) = ψ A (a) because the matrix is reduced. Finally, to say that this holds for all reduced matrices of S is equivalent to saying that the equation ϕ ≈ ψ holds in every A ∈ Alg∗S. The reader may have noticed that the same proof actually shows that the class of all algebra reducts of any class M of reduced matrices such that S is complete with respect to M generates the same variety KS .

If a, b ∈ θ then π(a) = π(b) and thus FiS π(a) = FiS π(b) , A/θ A/θ −1 −1 therefore π FiS π(a) = π FiS π(b) . But by construction we A/θ know that Cθ = π −1 ◦ FiS ◦ π; therefore we get Cθ (a) = Cθ (b). Thus we have ∼ proved that θ ∈ Con HA (θ). The second part of the statement comes directly from the construction. 11 HA (θ) is also a full model of S. To prove the last part of the Lemma, take θ1 , θ2 ∈ ConA and consider the natural projections π1 : A → A/θ1 and π2 : A → A/θ2 . If moreover θ1 ⊆ θ2 then the mapping j(a/θ1 ) = a/θ2 is an epimorphism from A/θ1 onto A/θ2 , and the following diagram A π1✲ A/θ1 π2 j ✲ ❄ A/θ2 commutes.

For any A, the ordered set FMod S A, is a complete lattice, and the Tarski operator is a lattice isomorphism between FMod S A, and Con AlgS A, ⊆ . Note that, although FMod S A is a subset of the complete lattice of all abstract logics over A, it need not be a sublattice; indeed, we do not have nice characterizations of the lattice operations in FMod S A , . The only thing we can say is that, as a consequence of the preceding results, given any collection {Li : i ∈ I} of full models of S on the same algebra A, its infimum in the lattice of full models of S can be obtained as the abstract logic projectively generated from A/θ, Fi S (A/θ) by the canonical projection of A onto A/θ, where ∼ θ = { Ω (Li ) : i ∈ I}.

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