Abstract
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
ℜ𝔞RCAγ ⊆ K ⊆ Scℜ𝔞CA5,
that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜ𝔞CAγ⊂Scℜ𝔞CAγ 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.
Citation
Robin Hirsch. "Relation algebra reducts of cylindric algebras and complete representations." J. Symbolic Logic 72 (2) 673 - 703, June 2007. https://doi.org/10.2178/jsl/1185803629
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