For an ordinal $α$, $\sf PEA_α$ denotes the class of polyadic equality algebras of dimension $α$. We show that for several classes of algebras that are reducts of $\PEA_ω$ whose signature contains all substitutions and finite cylindrifiers, if $\B$ is in such a class, and $\B$ is atomic, then for all $n<ω$, $\Nr_n\B$ is completely representable as a $\PEA_n$. Conversely, we show that for any $2<n<ω$, and any variety $\sf V$, between diagonal free cylindric algebras and quasipolyadic equality algebras of dimension $n$, the class of completely representable algebras in $\sf V$ is not elementary.
In this paper we give an alternative construction using Monk like algebras that are binary generated to show that the class of strongly representable atom structures is not elementary. The atom structures of such algebras are cylindric basis of relation algebras, both algebras are based on one graph such that both the relation and cylindric algebras are representable if and only if the chromatic number of the graph is infinite. We also relate the syntactic notion of algebras having a (complete) neat embedding property to the semantical notion of having various forms of (complete) relativized r…
We study different representation theorems for various reducts of Heyting polyadic algebras. Superamalgamation is proved for several (natural reducts) and our results are compared to the finitizability problem in classical algebraic logic dealing with cylindric and polyadic (Boolean algebras). We also prove several new neat embedding theorems, and obtain that the class of representable algebras based on (a generalized) Kripke semantics coincide with the class of algebras having the neat embedding property, that is those algebras that are subneat reducts of algebras having $ω$ extra dimensions.
Let n be finite >2. We show that any class between S\Nr_n\CA_{n+3} and RCA_n is not atom canonical, and any class containing the class of completely representable algebras and contained in S_c\Nr_n\CA_{n+3} is not elementary. We show that there is no finite variable universal axiomatization of many diagonal free reducts of representable cylindric algebras of dimension n, like the varieties of representable diagonal-free cylindric algebras and Halmos' polyadic algebras (without equality). We apply our hitherto obtained algebraic results to show that the omitting types theorem fails for finite v…
This paper is a survey of recent results and methods in (Tarskian) algebraic logic. We focus on cylindric algebras. Fix 2<n<ω. Rainbow constructions are used to solve problems on classes consisting of algebras having a neat embedding property substantially generalizing seminal results of Hodkinson as well as Hirsch and Hodkinson on atom-canonicity and complete representations, respectively. For proving non-atom-canonicity of infinitely many varieties approximating the variety of representable algebras of dimension n, so-called blow up and blur constructions are used. Rainbow constructions are …
(1) Let 1\leq k\leq ω. Call an atom structure αweakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra \A, such that \At\A=α, \A\in \Nr_n\CA_ω and for every algebra $\B$ based on this atom structure there exists k\in ω$, k\geq 1, such that \B\in \Nr_n\CA_{n+k}. (2) Let k\leq ω. Call an atom structure αk complete, if there exists \A such that \At\A=αand \A\in S_c\Nr_n\CA_{n+k}. (3) Let k\leq ω. Call an atom structure $α$ k neat if there exists \A such that \At\A=α, and …
We show that the class of representable substitution algebras is characterized by a set of universal first order sentences. In addition, it is shown that a necessary and sufficient condition for a substitution algebra to be representable is that it is embeddable in a substitution algebra in which elements are distinguished. Furthermore, conditions in terms of neat embeddings are shown to be equivalent to representability.
In this paper we investigate using the methodology of algebraic logic, deep algebraic results to prove three new omitting types theorems for finite variable fragments of first order logic. As a sample, we show that it T is an L_n theory and |T|=lambda, lambda a regular cardinal, if T admits elimination of quantifiers, then T omits < 2^λ many non isolated {\it maximal} types. This is basically a result of Shelah's restricted to L_n. that is not completely representable. We also show, using a rainbow construction for cylindric algebras, that the omitting types theorem fails for L_n even if we co…
Let L be a quantifier predicate logic. Let K be a class of algebras. We say that K is sensitive to L, if there is an algebra in K, that is L interpretable into an another algebra, and this latter algebra is elementary equivalent to an algebra not in K. (In particular, if L is L_{ω,ω}, this means that K is not elementary). We show that the class of neat reducts of every dimension is sensitive to quantifier free predicate logics with infinitary conjunctions; for finite dimensions, we do not need infinite conjunctions.
We view the neat reduct operator as a functor that lessens dimensions from CA_{α+ω} to CA_α for infinite ordinals α. We show that this functor has no right adjoint. Conversely for polyadic algebras, and several reducts thereof, like Sain's algebras, we show that the analagous functor is an equivalence.
In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatisation. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting types theorems, both negative and positive for finite variable fragments and / or modifications of first order logic. Negative results are obtained when we keep usual syntax and relativize models (so that they witness commutativity of quantifiers only locally) and positive ones are obtained when we weaken 'commutativity of quantifiers' in the syntax and relativi…
We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow constructions invented by Hirsch and Hodkinson. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. The paper has a survey character but it contains new approaches to old ones. We aspire to make our survey fairly comprehensive, at least in so …
In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
Using model theoretic techniques that proved that the class of $n$ neat reducts of $m$ dimensional cylindric algebras, $\Nr_n\CA_m$, is not elementary, we prove the same result for $\Ra\CA_k$, $k\geq 5$, and we show that $\Ra\CA_k\subset S_c\Ra\CA_k$ for all $k\geq 5$. Conversely, using the rainbow construction for cylindric algebra, we show that several classes of algebras, related to the class $\Nr_n\CA_m$, $n$ finite and $m$ arbitrary, are not elementary. Our results apply to many cylindric-like algebras, including Pinter's substitution algebras and Halmos' polyadic algebras with and withou…
Using constructions of Hirsch and Hodkinson, we show that the class of strongly atom structures for various cylindric-like algebras is not elementary. This applies to diagonal free reducts and polyadic algebras with and without equality. This follows from the simple observation, that the cylindric atom structures of dimension n, constructed by Hirsch and Hodkinson, are built from algebras that can be easily expanded to polyadic equality algebras, and that such expansions are generated by elements whose dimension sets <n.
We study what we call topological cylindric algebras and tense cylindric algebras defined for every ordinal $α$. The former are cylindric algebras of dimension $α$ expanded with $\sf S4$ modalities indexed by $α$. The semantics of representable topological algebras is induced by the interior operation relative to a topology defined on their bases. Tense cylindric algebras are cylindric algebras expanded by the modalities $F$(future) and $P$ (past) algebraising predicate temporal logic. We show for both tense and topological cylindric algebras of finite dimension $n>2$ that infinitely many vari…
We prove the result in the title. We infer, that unlike cylindric algebras, there is a first order axiomatization of the class of completely representable polyadic algebras of infinite dimension, though the one we obtain is infinite; in fact uncountable, but shares a single schema, stipulating that the (uncountably many)substitution operators are completely additive. Similar results are obtained for non commutative reducts of polyadic equality algebras of infinite dimensions, where we can drop complete additivity. However, it remains unknown to us whether there are atomic polyadic algebras of …
For any pair of ordinals $α<β$, $\sf CA_α$ denotes the class of cylindric algebras of dimension $α$, $\sf RCA_α$ denote the class of representable $\sf CA_α$s and $\sf Nr_αCA_β$ ($\sf Ra CA_β)$ denotes the class of $α$-neat reducts (relation algebra reducts) of $\sf CA_β$. We show that any class $\sf K$ such that $\sf RaCA_ω\subseteq \sf K\subseteq RaCA_5$, $\sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<ω$. It is also shown that any class $\sf K$ such that $\sf Nr_nCA_ω\cap {\sf CRCA}_n\subseteq {\sf K}\subseteq \mathbf{S}_c\sf Nr_nCA_{n+3}$, where $\sf CRCA_n$ is …
Topological cylindric algebras of dimension α, αany ordinal are cylindric algebras with dimension αexpanded with αS4 modalities. The S4 modalities in representable algebras are induced by a topology on the base of the representation of its cylindric reduct, that is not necessarily an Alexandrov topolgy. For α>2, the class of representable algebras is a variety that is not axiomatized by a finite schema, and in fact all complexity results on representations for cylindric algebras, proved by Andreka (concerning number of variables needed for axiomatizations) Hodkinson (on Sahlqvist axiomatizatio…
We show that there exists an atomic representable polyadic equality algebra of finite dimension n\geq 3, such that the cylindric reduct of its completion is not in SNr_n\CA_{n+4}, hence the result in the title. This solves an open problem in algebraic logic, though the values for k=n+1, n+2, n+3, is still, to the best of our knowlege unknown.
