Games in algebraic logic: axiomatisations and beyond

It is shown that the relevant logic R is complete with respect to squareincreasing, commutative, integral relation algebras. This contrasts with the representable case, where completeness is known to fail.

Algebra Universalis, 1995

We solve a problem of J6nsson by showing that the class Y/of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ~ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017

The algebra of binary relations provides union and composition as basic operators, with the empty set as neutral element for union and the identity relation as neutral element for composition. The basic algebra can be enriched with additional features. We consider the diversity relation, the full relation, intersection, set difference, projection, coprojection, converse, and transitive closure. It is customary to express boolean queries on binary relational structures as finite conjunctions of containments. We investigate which features are primitive in this setting, in the sense that omitting the feature would allow strictly less boolean queries to be expressible. Our main result is that, modulo a finite list of elementary interdependencies among the features, every feature is indeed primitive.

Journal of Symbolic Logic, 1999

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the ‘finite base property’ and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

Logical Foundations of Computer Science, 2021

In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [11]. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-andforth games.

The Journal of Symbolic Logic, 2021

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

Stanford Encyclopedia of Philosophy, publicado a, 2009

the algebra of logic were the most developed form of mathematical above all through Schröder's three volumes Vorlesungen über die Algebra der Logik (1890-1905). Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. In addition, in 1941, Alfred Tarski (1901-1983) in his paper "On the calculus of relations" returned to Peirce's relation algebra as presented in Schröder's Algebra der Logik. The tradition of the algebra of logic played a key role in the notion of Logic as Calculus as opposed to the notion of Logic as Universal Language. Beyond Tarski's algebra of relations, the influence of the algebraic tradition in logic can be found in other mathematical theories, such as category theory. However this influence lies outside the scope of this entry, which is divided into 10 sections.

We investigate in an algebraic setting the question in which logical languages the properties integral, permutational, and rigid of algebras of relations can be expressed.

Publications de l'Institut Mathematique, 2014

Contrary to the notion of a set or a tuple, a multiset is an unordered collection of elements which do not need to be different. As multisets are already widely used in combinatorics and computer science, the aim of this paper is to get on track to algebraic multiset theory. We consider generalizations of known results that hold for equivalence and order relations on sets and get several properties that are specific to multisets. Furthermore, we exemplify the novelty that brings this concept by showing that multisets are suitable to represent partial orders. Finally, after introducing the notion of an algebra on multisets, we prove that two algebras on multisets, whose root algebras are isomorphic, are in general not isomorphic.

Springer eBooks, 2018

Fragments of Tarski's relation algebra form the basis of many versatile graph and tree query languages including the regular path queries, XPath, and SPARQL. Surprisingly, however, a systematic study of the relative expressive power of relation algebra fragments on trees has not yet been undertaken. Our approach is to start from a basic fragment which only allows composition and union. We then study how the expressive power of the query language changes if we add diversity, converse, projections, coprojections, intersections, and/or difference, both for path queries and Boolean queries. For path queries, we found that adding intersection and difference yields more expressive power for some fragments, while adding one of the other operators always yields more expressive power. For Boolean queries, we obtain a similar picture for the relative expressive power, except for a few fragments where adding converse or projection yields no more expressive power. One challenging problem remains open, however, for both path and Boolean queries: does adding difference yields more expressive power to fragments containing at least diversity, coprojections, and intersection?