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In this paper we give an alternative construction using Monk like algebrasthat are binary generated to show that the class of strongly representable atomstructures is not elementary. The atom structures of such algebras arecylindric basis of relation algebras, both algebras are based on one graph suchthat both the relation and cylindric algebras are representable if and only ifthe chromatic number of the graph is infinite. We also relate the syntacticnotion of algebras having a (complete) neat embedding property to thesemantical notion of having various forms of (complete) relativizedrepresentations. Finally, we show that for n>5, the problemn as to whether afinite algebra is in the class SNr_3CA_6 is undecidable. In contrast, we showthat for a finite algebra of arbitary finite dimensions that embed into extradimensions of a another finite algebra, then this algebra have a finiterelativized representation. Finally we devise what we call neat games, for sucha game if \pe\ has a \ws \ on an atomic algebra \A in certain atomic game and\pa has a \ws in another atomic game, then such algebras are elementaryequivalent to neat reducts, but do not have relativized (local) completerepresenations. From such results, we infer that the omitting types theorem forfinite variable fragments fails even if we consider clique guarded semantics.The size of cliques are determined by the number of pebbles used by \pa\.

Strongly representable atom structures and neat embeddings