Relation algebra reducts of cylindric algebras and complete representations

Abstract

AbstractWe show, for any ordinalγ≥ 3, that the classℜaCAγis pseudo-elementary and has a recursively enumerable elementary theory. ScKdenotes the class of strong subalgebras of members of the classK. We devise games,Fn(3 ≤n≤ω),G, H, and show, for an atomic relation algebrawith countably many atoms, thatfor 3 ≤n<ω. We use these games to show, forγ> 5 and any classKof relation algebras satisfyingthatKis not closed under subalgebras and is not elementary. For infiniteγ, the inclusion ℜaCAγ⊂ScℜaCAγis strict.For infiniteγand for a countable relation algebrawe show thathas a complete representation if and only ifis atomic and ∃ has a winning strategy inF(At()) if and only ifis atomic and∈ScℜaCAγ.