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Amalgamation in relation algebras

Published online by Cambridge University Press:  12 March 2014

Maarten Marx
Maarten Marx*
Affiliation:
Department of Computing, Imperial College, 180 Queen's Gate, SW7 2BZ London, UK, E-mail: m.marx@doc.ic.ac.uk
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Extract

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].

Let K be a class of relational type algebras such that

(i) composition is associative,

(ii) K is a class of boolean algebras with operators, and

(iii) K contains the representable relation algebras RRA.

Then the equational theory of K is undecidable.

On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 479 - 484
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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