On the Algebraization of Some Gentzen Systems1

Let τ be a type of algebras without nullary fundamental operation symbols. An identity . φ ≈ ψ of type τ we call clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are non empty and identical. For a variety ν of type τ we denote by ν c the variety of type τ defined by all clone compatible identities from Id(ν). We call ν c the clone extension of ν. In this paper we describe identities, equational bases and minimal generics of all subvarieties of \( {\user1{{\mathcal{D}}}}^{c} \), where $$ {\user1{{\mathcal{D}}}} $$ is the variety of distributive lattices.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of the deductive interpolation property for axiomatic extensions of substructural logics and the amalgamation property for varieties of pointed residuated lattices, their equivalent algebraic semantics, is far less well understood, however. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule--algebraically, the commutativity law--in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly sixty varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly sixty axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly sixty varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

According to a result of Lee, varieties of pseudocomplemented distributive lattices form an ω+1 chain in which is the trivial variety and is the variety of Boolean algebras. In the present paper it is shown that the variety contains an almost universal subcategory B in which the members of Hom(B,B') associated with minimal prime ideals of B form a countably infinite set for any B,B' ∈ B. In particular, B3contains arbitrarily large algebras whose nontrivial endomorphisms form the countably infinite right zero semigroup. Our earlier results concerning categorical properties of varieties of pseudocomplemented distributive lattices show that no further reduction of the right zero count is possible.

One desirable property of a deductive system is that it should be easy to mechanize an efficient proof search. It is very difficult to search for a proof in a Hilbert system because there is no obvious connection between the formula and its proof. Proof search in the propositional calculus is easy and efficient with semantic tableaux and the equivalent (cut-free) Gentzen systems. However, as we shall see in the next chapter, the method of semantic tableaux in the predicate calculus becomes arbitrary and inefficient. The method of resolution, invented by J. A. Robinson in 1965, is frequently an efficient method for searching for a proof. In this section, we introduce resolution for the propositional calculus, though its advantages will not be apparent until it is extended to the predicate calculus.

In an earlier paper, we investigated for finite lattices a concept introduced by A. Slavik: LetA, B,andSbe sublattices of the latticeL,A∩B=S,A∪B=L. ThenL pastes AandBtogether overS, if every amalgamation ofAandBoverScontainsLas a sublattice. In this paper we extend this investigation to infinite lattices. We give several characterizations of pasting; one of them directly generalizes to the infinite case the characterization theorem of A. Day and J. Ježk. Our main result is that the variety of all modular lattices and the variety of all distributive lattices are closed under pasting.

On the Partially Ordered Monoid Generated by the Operators H, S, P, Ps on Classes of Algebras

By a result of K.B. Lee, the lattice of varieties of pseudo-complemented distributive lattices is the ω + 1 chainwhere B−1, B0, B1 are the varieties formed by all trivial, Boolean, and Stone algebras, respectively. General theorems on relative universality proved in the present paper imply that there is a proper class of non-isomorphic algebras in B3 with finite endomorphism monoids, while every infinite algebra from B2 has infinitely many endomorphisms. The variety B4 contains a proper class of non-isomorphic algebras with endomorphism monoids consisting of the identity and finitely many right zeros; on the other hand, any algebra in B3 with a finite endomorphism monoid of this type must be finite.

An equational identity of a given type τ involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identityδ: x ⋅ (y + z) = x ⋅ y + x ⋅ z has three variable symbols {x, y, z} and two operation symbols {+, ⋅}. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers 〈Z; +, ⋅〉 satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y ⋅ z) = (x + y) ⋅ (x + z) is not valid. By contrast, δis hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 〈2, 2, ⋯〉.

Natural dualities for varieties of distributive lattices with a quantifier

Equations in the theory of Q-distributive lattices

We investigate algebraically the fragment of the intuitionistic propositional calculus consisting of equivalence together with conjunction on the intuitionistic regularizations. We find that this fragment is strongly algebraizable with the equivalent algebraic semantics being the variety of equivalential algebras with an additional binary operation that can be interpreted as the meet on regular elements. We give a finite equational base for this variety, and investigate its properties, in particular the commutator. As applications, we prove that the fragment is hereditarily structurally complete and we describe the top of the lattice of its axiomatic extensions.

Direct consistency proof of Gentzen's system of natural deduction.

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

A κ-denotational semantics for map theory in ZFC + SI