Abstract
We investigate the class SRaCA n for 4 ≤ n < ω and survey some recent results. We see that RA n — the subalgebras of relation algebras with relational bases — is too weak, and that the class of relation algebras whose canonical extension has an n-dimensional cylindric basis is too strong to define the class. We introduce the notion of an n-dimensional hyperbasis and show that for any relation algebra A the canonical extension A + has such a hyperbasis if and only if A ∈ SRaCA n .
We introduce techniques that can be used to show that the hierarchies RA4 ⊃ RA5 ⊃... and SRaCA4 ⊃ SRaCA5 ⊃... are strict and each step is not finitely axiomatisable.
We outline a relativized semantics that characterises RA n and another one for the class of subalgebras of relation algebras with n-dimensional cylindric bases.
Research of first author partially supported by UK EPSRC grant GR/L85961
Research of second author partially supported by UK EPSRC grants GR/K54946 and GR/L85978.
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References
Andréka H., Monk J.D. and Németi I. (1991) Algebraic Logic. Colloq. Math. Soc. J. Bolyai. North-Holland, Amsterdam, 1991. Conference Proceedings, Budapest, 1989
Hirsch R. and Hodkinson I. (1997) Step by step — building representations in algebraic logic. Journal of Symbolic Logic, 62(1):225–279, March
Hirsch R. and Hodkinson I. (1998) Relation algebras with n-dimensional bases. submitted to Annals of Pure and Applied Logic
Hirsch R. and Hodkinson I. (1999) Relation algebras from cylindric algebras, I. Submitted to the Annals of Pure and Applied Logic
Hirsch R. and Hodkinson I. (1999) Relation algebras from cylindric algebras, II. Submitted to the Annals of Pure and Applied Logic
Hirsch R., Hodkinson I. and Maddux R. (1998) On the number of variables required in proofs. in preparation
Henkin L., Monk J.D. and Tarski A. (1971) Cylindric Algebras Part I. NorthHolland
Henkin L., Monk J.D. and Tarski A. (1985) Cylindric Algebras Part II. NorthHolland
Lyndon R. (1950) The representation of relational algebras. Annals of Mathematics, 51(3):707–729
Lyndon R. (1961) Relation algebras and projective geometries. Michigan Mathematics Journal, 8:207–210
Maddux R. (1978) Topics in Relation Algebra. PhD thesis, University of California, Berkeley
Maddux R. (1992) Relation algebras of every dimension. Journal of Symbolic Logic, 57(4):1213–1229
Monk J.D. (1964) On representable relation algebras. Michigan Mathematics Journal, 11:207–210
Stone M. (1936) The theory of representations for boolean algebras. Transactions of the American Mathematical Society, 40:37–111
Tarski A. and Givant S.R. (1987) A Formalization of Set Theory Without Variables. Number 41 in Colloquium Publications in Mathematics. American Mathematical Society, Providence, Rhode Island
Thompson R. (1993) Complete description of substitutions in cylindric algebras and other algebraic logics. In: Rauszer C. (Ed.) Algebraic methods in logic and in computer science, 28:327–342. Banach Center publications, Institute of mathematics, Polish Academy of Sciences
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Hirsch, R., Hodkinson, I. (2001). Connections Between Cylindric Algebras and Relation Algebras. In: Orłowska, E., Szałas, A. (eds) Relational Methods for Computer Science Applications. Studies in Fuzziness and Soft Computing, vol 65. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1828-4_14
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DOI: https://doi.org/10.1007/978-3-7908-1828-4_14
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