Relation algebra

Relation algebra

Relation algebra is a mathematical system that consists of a collection of binary relations over a specific domain D. These relations are both exclusive and complete, meaning that they divide the Cartesian product of D into distinct subsets. The module that deals with relation algebra provides various operations such as select, project, join, union, and diflf, as well as updating operations.From: Industrial and Engineering Applications of Artificial Intelligence and Expert Systems [2020], Simulation Systems [2019]

Reasoning About Relations Under Uncertainty

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Published in Don Potter, Manton Matthews, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2020

Bingning Dai, David A Bell, John G Hughes

Definition 2. Relation algebra. A relation algebraℜis a set of binary relations over some domain D, which are exclusive and complete, i.e., the relations partition the Cartesian product of D, D × D.

Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic

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Published in Journal of Applied Non-Classical Logics, 2020

Tarek Sayed Ahmed

The last result generalises to infinite dimensions replacing finite axiomatisation by axiomatised by a finite schema (Henkin et al., 1985; Hirsch & Sayed Ahmed, 2014). We consider relation algebras as algebras of the form , where is a Boolean algebra , ⌣ is a unary operation and; is a binary operation. A relation algebra is representable⇔ it is isomorphic to a subalgebra of the form , where X is an equivalence relation, is interpreted as the identity relation, ⌣ is the operation of forming converses, and; is interpreted as composition of relations. Following standard notation, denotes the class of (representable) relation algebras. The class is a discriminator variety that is finitely axiomatisable, cf. Hirsch and Hodkinson (2002, Definition 3.8, Theorems 3.19). All of the above classes of algebras are instances of s. The action of the non-Boolean operators in a completely additive (where operators distribute over arbitrary joins component-wise) atomic , is determined by their behaviour over the atoms, and this in turn is encoded by the atom structure of the algebra.

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Reasoning About Relations Under Uncertainty

Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic