Relation algebras from cylindric algebras, I

Accepted versio

Relation algebras with n-dimensional relational bases

Accepted versio

Strongly representable atom structures of relation algebras

Accepted versio

Axiomatizability of reducts of algebras of relations

Submitted versio

Strongly representable atom structures of cylindric algebras

Published versio

Canonical formulas for k-potent commutative, integral, residuated lattices

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for $k$-potent, commutative, integral, residuated lattices ($k$-$\mathsf{CIRL}$). We show that any subvariety of $k$-$\mathsf{CIRL}$ is axiomatised by canonical formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde

Algebras of partial functions

This thesis collects together four sets of results, produced by investigating modifications, in four distinct directions, of the following. Some set-theoretic operations on partial functions are chosen—composition and intersection are examples—and the class of algebras isomorphic to a collection of partial functions, equipped with those operations, is studied. Typical questions asked are whether the class is axiomatisable, or indeed finitely axiomatisable, in any fragment of first-order logic, what computational complexity classes its equational/quasiequational/first-order theories lie in, and whether it is decidable if a finite algebra is in the class. The first modification to the basic picture asks that the isomorphisms turn any existing suprema into unions and/or infima into intersections, and examines the class so obtained. For composition, intersection, and antidomain together, we show that the suprema and infima conditions are equivalent. We show the resulting class is axiomatisable by a universal-existential-universal sentence, but not axiomatisable by any existential-universal-existential theory. The second contribution concerns what happens when we demand partial functions on some finite base set. The finite representation property is essentially the assertion that this restriction that the base set be finite does not restrict the algebras themselves. For composition, intersection, domain, and range, plus many supersignatures, we prove the finite representation property. It follows that it is decidable whether a finite algebra is a member of the relevant class. The third set of results generalises from unary to ‘multiplace’ functions. For the signatures investigated, finite equational or quasiequational axiomatisations are obtained; similarly when the functions are constrained to be injective. The finite representation property follows. The equational theories are shown to be coNP-complete. In the last section we consider operations that may only be partial. For most signatures the relevant class is found to be recursively, but not finitely, axiomatisable. For others, finite axiomatisations are provided

Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness

We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary)