Neat reducts and amalgamation in retrospect, a survey of results and some methods Part I: Results on neat reducts
Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the (seemingly very innocent) fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new unpublished results (answering long-standing open problems in the field) concerning neat reducts and amalgamation are given. (Theorems 11, 13, 19 and 31-38 are such). Several counterexamples which convey the gist of techniques used in this area are presented two of which are new (Theorem 19, Theorem 38). It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [54] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. This paper appears in two parts. The present first part contains results on neat reducts. The second part contains results relating the notion of neat embeddings to various amalgamation properties.1