Strongly representable atom structures and neat embeddings

TL;DR

This paper explores the properties of strongly representable atom structures, showing their non-elementarity, and investigates the complexity of embedding problems and the failure of the omitting types theorem in finite variable fragments.

Contribution

It introduces an alternative construction for strongly representable atom structures, relates syntactic and semantic notions of neat embeddings, and establishes undecidability results for certain algebra classes.

Findings

Strongly representable atom structures are not elementary.

Embedding problems for finite algebras are undecidable beyond certain dimensions.

Omitting types theorem fails for finite variable fragments with clique guarded semantics.

Abstract

In this paper we give an alternative construction using Monk like algebras that are binary generated to show that the class of strongly representable atom structures is not elementary. The atom structures of such algebras are cylindric basis of relation algebras, both algebras are based on one graph such that both the relation and cylindric algebras are representable if and only if the chromatic number of the graph is infinite. We also relate the syntactic notion of algebras having a (complete) neat embedding property to the semantical notion of having various forms of (complete) relativized representations. Finally, we show that for n>5, the problemn as to whether a finite algebra is in the class SNr_3CA_6 is undecidable. In contrast, we show that for a finite algebra of arbitary finite dimensions that embed into extra dimensions of a another finite algebra, then this algebra have a finite relativized representation. Finally we devise what we call neat games, for such a 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 elementary equivalent to neat reducts, but do not have relativized (local) complete represenations. From such results, we infer that the omitting types theorem for finite variable fragments fails even if we consider clique guarded semantics. The size of cliques are determined by the number of pebbles used by \pa\.