Journal of Symbolic Logic

Relation algebra reducts of cylindric algebras and complete representations

Robin Hirsch

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We show, for any ordinal γ ≥ 3, that the class ℜ𝔞CAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fn (3≤ n≤ω), G, H, and show, for an atomic relation algebra 𝒜 with countably many atoms, that

  • ∃ has a winning strategy in Fω(At(𝒜))⋔ 𝒜∈Scℜ𝔞CAω,
  • ∃ has a winning strategy in Fn(At(𝒜)) ⇐ 𝒜∈Scℜ𝔞CAn,
  • ∃ has a winning strategy in G(At(𝒜)) ⇐ 𝒜∈ℜ𝔞CAω,
  • ∃ has a winning strategy in H(At(𝒜))→𝒜∈ℜ𝔞RCAω
for 3≤ n < ω. We use these games to show, for γ≥ 5 and any class K of relation algebras satisfying

ℜ𝔞RCAγ ⊆ K ⊆ Scℜ𝔞CA5,

that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜ𝔞CAγScℜ𝔞CAγ is strict. For infinite γ and for a countable relation algebra 𝒜 we show that 𝒜 has a complete representation if and only if 𝒜 is atomic and ∃ has a winning strategy in F(At(𝒜)) if and only if 𝒜 is atomic and 𝒜∈Scℜ𝔞CAγ.

Article information

J. Symbolic Logic, Volume 72, Issue 2 (2007), 673-703.

First available in Project Euclid: 30 July 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G15: Cylindric and polyadic algebras; relation algebras

relation algebra cylindric algebra complete representation


Hirsch, Robin. Relation algebra reducts of cylindric algebras and complete representations. J. Symbolic Logic 72 (2007), no. 2, 673--703. doi:10.2178/jsl/1185803629.

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