Epimorphisms in cylindric algebras and definability in finite variable logic

Abstract

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of n-dimensional cylindric algebras and the class of representable algebras in CAn for finite n > 1, solving Problem 10 of "Cylindric Set Algebras", by Henkin, et al. for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is greater than 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several theorems in the literature concerning injective algebras and definability of polyadic operations in CAn are best possible. We raise several open problems.

Finite-Variable Logics Do Not Have Weak Beth Definability Property

“…Thus the weak Beth definability property fails for a wide variety of logics, from the restricted n-variable fragment with finite models only, to L n ∞,ω . The variant of L n in which we allow only models of size ≤ n + 1 has the strong Beth definability property, for all n, this is proved in [2]. Another variant of L n that has the strong Beth definability property is when we allow models of all sizes but in a model truth is defined by using only a set of selected (so-called admissible) evaluations of the variables (a generalized model then is a pair consisting of a model in the usual sense and this set of admissible evaluations).…”

“…Failure of Beth definability property for the finite variable fragments was first proved in 1983 [3] (for all n ≥ 2) by showing that epimorphisms are not surjective in finite-dimensional cylindric algebras, see [2]. That proof, translated to logic, relies inherently on the fact that the implicit definition it uses is not satisfiable in each model of the theory.…”

Distances Between Formal Theories

“…The idea is to count the minimum number of axioms that are needed to be added or "removed" to get from one theory to the other. 1 We prove that the axiomatic distance between theories formulated in the same language must be ≤3. See Proposition 3.11 on Page 642.…”

“…In a future algebra-oriented article, we plan to discuss in detail the correspondence between our logical definition herein and the above algebraic idea. We note that it happens quite often that one obtains interesting results in mathematical logic by using algebraic tools, e.g., [17], [7], [1], [9], and [22]. We start with a quick review for the notions of logic that we are going to use.…”

The polyadic generalization of the Boolean axiomatization of fields of sets

“…The references [1], [5], [6], [7], [12], [14], [15], [16] are related indirectly to the topic or are related to the applications.…”