The class of infinite dimensional neat reducts of quasi-polyadic algebras is not axiomatizable

Abstract

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in Kβ is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Note on Neat Reducts

“…This confirms a conjecture of Németi formulated in [14] and solves problem 4.4 in [9]. In [2] this result is extended to Halmos' quasipolyadic algebras and Pinter's Substitution algebras, but again only for the lowest values of α namely when α = 2. Here we extend the result in [2] to arbitary finite dimensions.…”

“…We show that the class of n dimensional neat reducts of algebras in K m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2].…”

On the representability of neatly embeddable CA’s by cylindric relativized algebras

“…There are some remarkable connections between our topics and the following references: [3], [4], [9], [10], [11], [12], [13].…”

Some results about neat reducts

“…The closure of the class of neat reducts for other algebras under forming (elementary) subalgebras is investigated in [2], [8], [10].…”

“…This is not the end of the story; in fact, this is where the fun begins. A new unexpected viewpoint can yield dividends, and indeed the notion of neat reducts has been revived lately, to mention a few references: [3], [4], [2], [5], [1], [31], [53], [20], [23], [19], and [7]. In this paper, we survey (briefly) such results on neat reducts, putting them in a wider perspective.…”