What is modal logic

Publisher Summary This chapter focuses on the concept of Galois connexions. The researches in algebraic linguistics have been performed in the frame of a particular Galois connexion, in which the objects are the words of a vocabulary, V, and the set of the properties is the set of the contexts that accept these words. Vocabulary V is, however, formed by a finite of words. The chapter highlights that if A is a generating kernel for the closed set φ ( X ), then the set formed by taking one element of every family contained in A is also a generator for φ ( X ). Such a generator is called the minimal generator of the set φ( X ) with respect to the kernel A . The chapter also discusses the properties of the operators G , ω , and φ .

On Galois Connexions

Graph Structure and Monadic Second-Order Logic: Overview

Representable posets

For representation by partial functions in the signature with intersection, composition and antidomain, we show that a representation is meet complete if and only if it is join complete. We show that a representation is complete if and only if it is atomic, but that not all atomic representable algebras are completely representable. We show that the class of completely representable algebras is not axiomatisable by any existential-universal-existential first-order theory. By giving an explicit representation, we show that the completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal sentence.

Complete representation by partial functions for composition, intersection and anti-domain

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be the intersection of jCRL and mCRL. We prove CRL is a strict subset of biCRL which is a strict subset of both jCRL and mCRL. Let L be a DL. Then L is in mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L is in jCRL iff L has a distinguishing set of completely prime filters, and L is in CRL iff L has a distinguishing set of complete, completely prime filters. Each of the classes above is shown to be pseudo-elementary and hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.

Completely Representable Lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be ${{\bf jCRL} \cap {\bf mCRL}}$ . We prove

$${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$$

where the marked inclusions are proper.

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Completely representable lattices

No finite axiomatizations for posets embeddable into distributive lattices

A poset is $(\omega,C)$-representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of $(\omega,C)$-representable posets cannot be axiomatized in first order logic using the standard language of posets. We generalize this result to $(\alpha,\beta)$-representable posets for certain values of $\alpha$ and $\beta$.

Rob Egrot

Papers

Representable posets

Completely Representable Lattices

Completely representable lattices

No finite axiomatizations for posets embeddable into distributive lattices

Non-elementary classes of representable posets