## COMPLEXITY OF EQUATIONS VALID IN ALGEBRAS OF RELATIONS -- Part II: Finite axiomatizations.

Citations: | 17 - 2 self |

### BibTeX

@MISC{Andréka_complexityof,

author = {Hajnal Andréka},

title = {COMPLEXITY OF EQUATIONS VALID IN ALGEBRAS OF RELATIONS -- Part II: Finite axiomatizations.},

year = {}

}

### OpenURL

### 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, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn 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 ! !. Completely analogous statement holds for the case n !. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...