Neat Embeddings and Amalgamation

We present a property of neat reducts commuting with forming subalgebras as a definability condition. The purpose of this paper is to relate results on neat reducts, a notion particular to cylindric algebras to results on the “more universal” strong amalgamation property in a very general setting. It is known [7], [8] and [6] that amalgamation properties in a class of algebras correspond to interpolation and definability properties in the corresponding logic. Pigozzi [11] is a milestone for working out such eqiuvalences for cylindric algebras, cf. [9]. We follow the notation of [3]. Neat reducts [3] [2.6.28] is an old venerable notion in algebraic logic that is invented by Leon Henkin back in the fifties. Let CAβ stand for the class of cylindric algebras of dimension β. Let α < β. A neat α reduct of a CAβ A say, is an α-dimensional cylindric algebra obtained from A by keeping only the α-dimensional elements and discarding those operations indexed by β\α. More precisely: