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Let 2 < n < ω. Then CAn denotes the class of cylindric algebras of dimension n, RCAn denotes the class of representable CAns, CRCAn denotes the class of completely representable CAns, and NrnCAω(⊆ CAn) denotes the class of n-neat reducts of CAωs. The elementary closure of the class CRCAns (Kn) and the nonelementary class At(NrnCAω) are characterized using two-player zero-sum games, where At is the operator of forming atom structures. It is shown that Kn is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ScNrnCAω where Sc is the operation of forming complete subalgebras. For any class L such that AtNrnCAω ⊆ L ⊆ AtKn, it is proved that SPCmL = RCAn, where Cm is the dual operator to At; that of forming complex algebras. It is also shown that any class K between CRCAn ∩ SdNrnCAω and ScNrnCAn+3 is not first order definable, where Sd is the operation of forming dense subalgebras, and that for any 2 < n < m, any l ≥ n + 3 any any class K such that At(NrnCAm ∩ CRCAn) ⊆ K ⊆ AtScNrnCAl, K is not not first order definable either. [ABSTRACT FROM AUTHOR]

Fix a finite ordinal n ≥ 3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, and the class CRRA of completely representable RAs are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety V between polyadic algebras of dimension n and diagonal free CAns. We show that that the class of completely and strongly representable algebras in V is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class CRRA is not closed under =1,ω. In contrast, we show that given α ≥ ω, and an atomic A 2 PEAα, then for any n < !, NrnA is a completely representable PEAn. We show that for any α ≥ ω, the class of completely representable algebras in certain reducts of PAαs, that happen to be varieties, is elementary. We show that for α ≥ ω, the the class of polyadic-cylindric algebras dimension α, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension n and PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second. [ABSTRACT FROM AUTHOR]