Nonfinite axiomatizability results for cylindric and relation algebras

Abstract

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

Connections between Relation Algebras and Cylindric Algebras

“…In a number of publications, including notably [5,6] and the survey [7], Maddux gave a new way of constructing cylindric algebras of any dimension from atomic relation algebras, using sets of networks called cylindric bases. We will continue to simplify things by restricting consideration to finite simple relation algebras.…”

“…The question of connections between algebras of different arities (and perhaps varying in other features too) is of interest in its own right. It has a distinguished history, including work of Monk [8] and Maddux [5][6][7]. It raises intricate technical challenges.…”

Residuated Kleene Algebras

“…The construction in [7] is rather involved, since we needed dense algebras (where x ≤ x;x is valid) so that they can be applied to relevance logic. A simpler construction is available from [11] that has been used to show nonfinite axiomatizability of the equational theories of some residuated algebras in [12]. The Kleene star operation was not considered in [12], but it can be easily added to the signature.…”

“…We recall the main features of the algebras A n = (A n , −, ·, +, ;, , 0, 1 , 1) of [11]. A n has the following atoms (minimal, nonzero elements): identity 1 , q i for 1 ≤ i ≤ m, and p j for 1 ≤ j ≤ n with m = 3 · n!.…”

“…It is shown in [11] that any nonprincipal ultraproduct A of the A n 's is representable. By the above definition of Kleene star in A n , it follows that we can define * in the same way in A and that the representation respects * as well.…”