A Stone-type representation theorem for algebras of relations of higher rank

H. Andr{éka, R. J. Thompson
1988 Transactions of the American Mathematical Society  
The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of n-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on n-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The
more » ... son theorem we prove here gives a finite set of equations axiomatizing the class of algebras of n-ary relations (for every ordinal n). The (Resek-Thompson) theorem we are going to prove here is a "geometric" representation theorem for cylindric algebras. It provides an apparently satisfactory positive solution to the representation problem of cylindric algebras (summed up, e.g., in the introduction of [HMTI] and in, e.g., Henkin-Monk [74]). The theorem represents every "abstract" algebra satisfying the cylindric axioms (eight schemes of equations; cf. the remarks on the choice of the axioms at the end of the paper) by a "concrete" algebra of sets of sequences. The representing algebra is concrete in the sense that we do not have to know the operations of the algebra, it is enough to know its elements. I.e. if we know the elements of the algebra, we can "compute" the operations on them by using their concrete set theoretic structure. (This is similar to the Boolean case where if x, y are elements of a concrete algebra 93 then their meet must be the set theoretic x fl y independently of the choice of 93. Already in the Boolean case we have to know the greatest element of 93 in order to be able to compute the complement -x of x in 93.) The first version of the theorem was obtained by Diane Resek and is proved as Theorem 5.27 on p. 285 of Resek [75]. Resek's result is also announced in [HMTII, p. vi, p. 101 (item 3.2.88)] and Henkin-Resek [75, Theorem 4.3], and is mentioned, e.g., in Maddux [82] preceding Problem 5.21; but no proof has appeared in print for this important theorem so far (for reasons indicated below). Using the techniques of Thompson [79], Richard J. Thompson generalized Resek's theorem to the form in which it appears below. Thompson's result is (partially) quoted in [HMTII, 3.2.88] without proof, and otherwise is unpublished. Thompson's proof is of a proof theoretic nature and proves more than the theorem stated below. Further
doi:10.1090/s0002-9947-1988-0961607-5 fatcat:quremgfvkze2rgldou6pyhvqwy