Relation algebra reducts of cylindric algebras and complete representations

Abstract

We show, for any ordinal γ ≥ 3, that the class ℜ𝔞CA_γ is pseudo-elementary and has a recursively enumerable elementary theory. S_cK 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 F^ω(At(𝒜))⋔ 𝒜∈S_cℜ𝔞CA_ω,

• ∃ has a winning strategy in Fⁿ(At(𝒜)) ⇐ 𝒜∈S_cℜ𝔞CA_n,

• ∃ has a winning strategy in G(At(𝒜)) ⇐ 𝒜∈ℜ𝔞CA_ω,

• ∃ has a winning strategy in H(At(𝒜))→𝒜∈ℜ𝔞RCA_ω

for 3≤ n < ω. We use these games to show, for γ≥ 5 and any class K of relation algebras satisfying

ℜ𝔞RCA_γ ⊆ K ⊆ S_cℜ𝔞CA₅,

that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜ𝔞CA_γ⊂S_cℜ𝔞CA_γ 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∈S_cℜ𝔞CA_γ.