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Register a new userGeometrical representation theorems for cylindric-type algebras
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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 can also be viewed as multi-modal logics. The proofs herein are combinatorial and we also use some techniques from game theory.
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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 $alpha$-neat reducts (relation algebra reducts) of $sf CA_beta$. We show that any class $sf K$ such that $sf RaCA_omega subseteq sf Ksubseteq RaCA_5$, $sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<omega$. It is also shown that any class $sf K$ such that $sf Nr_nCA_omega cap {sf CRCA}_nsubseteq {sf K}subseteq mathbf{S}_csf Nr_nCA_{n+3}$, where $sf CRCA_n$ is the class of completely representable $sf CA_n$s, and $mathbf{S}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $sf K$ such that $mathbf{S}_dsf Ra CA_omega subseteq {sf K}subseteq mathbf{S}_csf RaCA_5$ is not elementary. It remains to be seen whether there exist elementary classes between $sf RaCA_omega$ and $mathbf{S}_dsf RCA_{omega}$. In particular, for $mgeq n+3$, the classes $sf Nr_nCA_m$, $sf CRCA_n$, $mathbf{S}_dsf Nr_nCA_m$, where $mathbf{S}_d$ is the operation of forming dense subalgebras are not first order definable.
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 variables briefly $sf TopL_{alpha}$. This is a sequel to cite{3}; the second part on modal cylindric algebras, where we study algebraically other properties of $sf TopL_{alpha}$. Modal cylindric algebras are cylindric algebras of infinite dimension expanded with unary modalities inheriting their semantics from a unimodal logic $sf L$ such as $sf K5$ or $sf S4$. Using the methodology of algebraic logic, we study topological (when $sf L=S4$), in symbols $sf TCA_{alpha}$. We study completeness and omitting types $sf OTT$s for $sf TopL_{omega}$ and $sf TenL_{omega}$, by proving several representability results for locally finite such algebras. Furthermore, we study the notion of atom-canonicity for both ${sf TCA}_{n}$ and ${sf TenL}_n$, a well known persistence property in modal logic, in connection to $sf OTT$ for ${sf TopL}_n$ and ${sf TeLCA}_n$, respectively. We study representability, omitting types, interpolation and complexity isssues (such as undecidability) for topological cylindric algebras. In a sequel to this paper, we introduce temporal cyindric algebras and point out the way how to amalgamate algebras of space (topological algebars) and algebras of time (temporal algebras) forming topological-temporal cylindric algebras that lend themselves to encompassing spacetime gemetries, in a purely algebraic manner.
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 $mgeq n+2$.
Boolean-type algebra (BTA) is investigated. A BTA is decomposed into Boolean-type lattice (BTL) and a complementation algebra (CA). When the object set is finite, the matrix expressions of BTL and CA (and then BTA) are presented. The construction and certain properties of BTAs are investigated via their matrix expression, including the homomorphism and isomorphism, etc. Then the product/decomposition of BTLs are considered. A necessary and sufficient condition for decomposition of BTA is obtained. Finally, a universal generator is provided for arbitrary finite universal algebras.
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Grobner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Grobner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss cite{Wemyss}, related to the geometric argument of Toda cite{T}. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree $ngeq 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class ${cal P}_n$ of potential algebras with homogeneous potential of degree $n+1geq 4$, the minimal Hilbert series is $H_n=frac{1}{1-2t+2t^n-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants.
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