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Let $2<n<mleq omega$. Let $CA_n$ denote the class of cylindric algebras of dimension $n$ and $RCA_n$ denote the class of representable $CA_n$s. We say that $Ain RCA_n$ is representable up to $m$ if $CmAtA$ has an $m$-square representation. An $m$ square represenation is locally relativized represenation that is classical locally only on so called $m$-squares. Roughly if we zoom in by a movable window to an $m$ square representation, there will become a point determinded and depending on $m$ where we mistake the $m$ square-representation for a genuine classical one. When we zoom out the non-representable part gets more exposed. For $2<n<m<lleq omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety $RCA_n$ is a limiting case coinciding with $CA_n$s having $omega$-square representations. Let $RCA_n^m$ be the class of algebras representable up to $m$. We show that $RCA_n^{m+1}subsetneq bold RCA_n^m$ for $mgeq n+2$.
In cite{CGH15} we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of cite{CGH15} we answer Problem 2 from there by characterising the perfect lattices that are
For any pair of ordinals $alpha<beta$, $sf CA_alpha$ denotes the class of cylindric algebras of dimension $alpha$, $sf RCA_{alpha}$ denote the class of representable $sf CA_alpha$s and $sf Nr_alpha CA_beta$ ($sf Ra CA_beta)$ denotes the class of $alp
In this paper, we give new proofs of the celebrated Andreka-Resek-Thompson representability results of certain axiomatized cylindric-like algebras. Such representability results provide completeness theorems for variants of first order logic, that ca
Let $alpha$ be an arbritary ordinal, and $2<n<omega$. In cite{3} accepted for publication in Quaestiones Mathematicae, we studied using algebraic logic, interpolation, amalgamation using $alpha$ many variables for topological logic with $alpha$ many
For an ordinal $alpha$, $sf PEA_{alpha}$ denotes the class of polyadic equality algebras of dimension $alpha$. We show that for several classes of algebras that are reducts of $PEA_{omega}$ whose signature contains all substitutions and finite cylind