Amalgamation properties and interpolation theorems for equational theories

DEFINITION 0.1. Let Fx, F2, F be subsets of a boolean algebra B. Then the (F1, F2, F)-Separation Principle, written Sep(F~, F2, F) for short, is the following: whenever aEF1, bEF2 and a ^ b = O , there is c~F with a<.c and cAb=O. The (F1, F2, F)-Interpolation Principle, written Int(Ft , F2, F) for short, is the following: whenever a~F1, b~F2 and a<.b there is ceF such that a<~c<~b. Clearly Int (F~, /'2, / ') is just Sep (F l, F 2 , F) where F 2 = { b:b ~F2}, so that the two notions are coextensive. Which one to use is a matter of taste, convenience and history. In practice B is often the Lindenbaum algebra of formulas reduced modulo some theory: then we replace each equivalence class [~p] by its representative formula q~, the ordering [~0]<,. [O] of the algebra becomes a valid implication ~0-,% and O denotes falsity. We also abuse notation mildly by using F~, etc., for classes of formulas. Several standard examples of separation principles occur in Recursion Theory, Descriptive Set Theory and Model Theory: however, they are usually symmetric in the sense that F1 = F2, and strict in the sense that F~ ca F2 = F. In contrast, we shall often want to dispense with these assumptions, and this is why we phrase the definition with three parameters as opposed to the usual one. The best known interpolation theorem in Model Theory is probably Craig's Theorem: this can be summarised as In t (E 2, U 2, F) where F is the set of all first order sentences in a language L and E z [respectively U 2] is the set of existential [universal] second-order sentences over L. We shall, however, be interested only in first order formulas, of low quantifier complexity, and so we begin by looking at an interpolation theorem for such formulas, Herbrand's Theorem.