Abstract
The theory of cylindric algebras was introduced byTarski in the fifties of the twentieth century, and its intensive study was further pursued by pioneers such as Henkin and Monk and, by the Hungarian mathematicians Andreka, Nemeti and Sain, and many of their students; to name only a few: Madarasz, Marx, Kurucz, Simon, Mikulas, and Sagi and many others outside Hungary including the author of this paper. Here we introduce and investigate new notions of representability for cylindric algebras and investigate various connections between such notions. Let 2 < n <= l < m <= omega. Let CA(n) denote the variety of cylindric algebras of dimension n and let RCA(n) denote the variety of representable CA(n)s. We say that an atomic algebra 2l is an element of CA(n) has the complex neat embedding property up to l and m if 2l is an element of RCA(n) boolean AND Nr(n)CA(l) and CmAt2l is an element of SNr(n)CA(m). Fixing the prarameters l at the value n, this is a measure of howmuch the algebra is representable. The yardstick is how far can its Dedekind-MacNeille completion be dilated, that is to say, counting the number of more extra dimensions its Dedekind-MacNeille completion neatly embeds into. If 2l, B is an element of RCA(n) are atomic, CmAtB is an element of SNr(n)CA(l) and CmAt2l is an element of SNr(n)CA(m), then we say that 2l is more representable than B. When m = omega, we say that 2l is strongly representable; this is the maximum degree of representability; the algebra in question cannot be 'more representable' than that. In this case the atom structure of 2l, namely At2l, is strongly representable in the sense of Hirsch and Hodkinson. This notion gives an infinite potential spectrum of 'degrees' of representability. In this connection, we exhibit various atomic algebras in RCA(n) boolean AND Nr(n)CA(l) that do no not have the complex neat embedding property for infinitely many values of l and m. It is known that the class of Kripke frames Str(RCA(n)) = {F : CmF is an element of RCA(n)} is not elementary. From this it follows that there is some n < m < omega such that Str(SNr(n)CA(m)) = {F : CmF is an element of SNr(n)CA(m)} is not elementary. Replacing S by S-c (forming complete subalgebras), S-d (forming dense subalgebras) and I (forming isomorphic copies), respectively, we show that for any O is an element of{S-c, S-d, I}, the class of frames Str(ONr(n)CA(n+3)) = {F : CmF is an element of ONr(n)CA(n+3)} is not elementary. Metalogical applications are given to n-variable fragments of first-order logic endowed with so-called clique guarded semantics. The last semantics capture the new notions of representations introduced and studied in this paper.