Theory Of Relation Algebras Research Articles

Relations Between Spatial Calculi About Directions and Orientations

Qualitative spatial descriptions characterize essential properties of spatial objects or configurations by relying on relative comparisons rather than measuring. Typically, in qualitative approaches only relatively coarse distinctions between configurations are made. Qualitative spatial knowledge can be used to represent incomplete and underdetermined knowledge in a systematic way. This is especially useful if the task is to describe features of classes of configurations rather than individual configurations. Although reasoning with them is generally NP-hard, relative directions are important because they play a key role in human spatial descriptions and there are several approaches how to represent them using qualitative methods. In these approaches directions between spatial locations can be expressed as constraints over infinite domains, e.g. the Euclidean plane. The theory of relation algebras has been successfully applied to this field. Viewing relation algebras as universal algebras and applying and modifying standard tools from universal algebra in this work, we (re)define notions of qualitative constraint calculus, of homomorphism between calculi, and of quotient of calculi. Based on this method we derive important properties for spatial calculi from corresponding properties of related calculi. From a conceptual point of view these formal mappings between calculi are a means to translate between different granularities.

Mitsch’s order and inclusion for binary relations and partitions

Mitsch’s natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described.

Groups and Algebras of Binary Relations

AbstractIn 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jónsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jónsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms—called coset relation algebras—that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jónsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

Products of modal logics, part 3: Products of modal and temporal logics .

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra $ \frak U $ an element x satisfying the condition (a) $ 0 \neq x; 1; \breve{x} + \breve{x}; 1; x \leq 1^{'} $ must be an atom of $ \frak U $ . It follows that x must also be an atom in every simple extension of $ \frak U $ . Andreka, Jonsson and Nemeti [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.¶ The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jonsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.¶ Andreka, Jonsson, Nemeti [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”.

A perspective on the theory of relation algebras

A perspective on the theory of relation algebras