Noncommutative cylindric algebras and relativizations of cylindric algebras
نویسنده
چکیده
We obtain the class N A of noncommutative cylindric algebras from the class CA of cylindric algebras by weakening the axiom C 4 of commuta-tivity of cylindrifications (to C * 4 , see below, and we obtain N CA from CA by omitting C 4 completely). Some motivation for studying noncommutative cylindric algebras: Noncommutative cylindric algebras (N A's) have the same " substi-tutional structure " as cylindric algebras (CA's from now on), where sub-stitutional structure refers to the equational behaviour of the substitution operations, the s i j 's. For certain technical reasons, N A's turn out to be useful in the study of the substitutional structure of CA's. Cf. Thm. 3 below. In a sense, N A's have a nicer representation theory than CA's. Namely, N A + M GR admits a geometric characterization in terms of certain " concrete " algebras (of sets of sequences), while the corresponding result for CA + M GR is only representation result but not a characterization result (some of the concrete algebras are not CA's). It is not clear to the present author how to obtain a similarly nice characterization (" if and only if " – type representation) result for CA + " some finite schema ". Cf. the remarks between Thm.s 6 and 7 here. A third motivation for N A's comes from the theory of relativizations of CA's extensively investigated in A fourth motivation comes from a strongly related development in relation algebra theory pursued e.g. in Maddux's thesis, Tarski-Givant [12], [9]-[10]. (E.g. if we take the CA version of the symmetric relativizations
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