Complexity of equations valid in algebras of relations part I: Strong non-finitizability

Abstract

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce and Schröder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCA n of cylindric algebras of n-ary relations, RPEA n of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCA n has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. A completely analogous statement holds for the case n ⩾ ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous non-finitizability properties of the larger varieties SNr nCA n + k . We prove that the complementation-free (i.e. positive) subreducts of RCA n do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCA n . We look at all the possible reducts of RCA n and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEA n and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment L n of first order logic. The reason for this is that RCA n and RPEA n are the natural algebraic counterparts of L n while the varieties SNr nCA n + k are in connection with the proof theory of L n . This paper appears in two parts. This is the first part, it contains the non-finite axiomatizability results. The second part contains finite axiomatizability results together with a figure summarizing the results in this area and the problems left open.