### The decision problem for the logic of predicates and of operations

Yu. Sh. Gurevich
1969 Algebra and Logic
Gurevich UDC518.5 INTR OD UC TI ON Most work on the decision problem for the logic of predicates belongs to the class of preliminary formulas, posed in a way limited to prefixes and predicate variables. These investigations have recently, in many respects, been completed. Corresponding results are introduced at the beginning of the second chapter of this work in extensive detail. It turns out that that part of these results can be anticipated from simple algebraic considerations and that the
more » ... uation is far more general, at any rate for any theory of first order. The first chapter of our work is dedicated to this. The content of the second chapter essentially generalizes the above-mentioned results on the decision problem for the logic of predicates to the case where there are assumed variables of operation. The second chapter can be considered, also, as an illustration of the first. The more difficult of the special results of the second chapter have been considered by us separately (see [3]). In distinction to the case of absence of operations, adjoining the equality sign in the case of available operations materially alters the picture. Our work is comparable with [4] for instance. In the second chapter, we consider only formulas without the equality sign. Clarification of the connections between the results of Chapter 2 and earlier results of other authors we postpone to §4 of Chapter 2. CHAPTER TIGHT SETS AND THE DECISION PROBLEM Tight Partially Ordered Sets This section is algebraic in character. In particular, as is acceptable and convenient in contemporary general algebra, we shall use the same symbol to denote a partially ordered set, i.e., a definite model, and the fundamental set of this model. Analogously, the concept "subset of a partially ordered set" will have two meanings. One meaning is submodel, the other -subset of the basic set. Instead of "partially ordered set" we shall abbreviate "p.o. set." All information needed from the theory of partial ordering can be found in §4 of Chapter 1 of the book [6]. Definition 1. (Definition of tightness). A p.o. set M is called tight if each sequence {x~} ~co of elements of M contains an increasing (not necessarily strictly increasing) subsequenee. Here u) denotes, as usual, the natural order in the set of all natural numbers. Obviously, if M is a tight p.o. set, then: THEOREM 1. /vf cannot contain an infinite subset of pairw~se noncomparable elements and THEOREM 2. ?¢I satisfies the descending chain condition. Conversely, if a p.o. set satisfies the conditions of Theorems 1 and 2, then it is tight. We shall not use here, and hence will not show, this converse assertion.