On completions of algebras in SNr_nCA_{n+k}, n\geq 3, k\geq 1

We give a sufficient condition that implies that SNr_nCA_{n+k}, n\geq 3, k\geq 3 (both finite)is not closed under completions. We compare this condition to existing results in literature

Completeness and interpolation for intuitionistic infinitary predicate logic, in connection to finitizing the class of representable Heyting polyadic algebras

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 $\omega$ extra dimensions.Comment: arXiv admin note: text overlap with arXiv:1304.0707, arXiv:1304.114

Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras

We study what we call topological cylindric algebras and tense cylindric algebras defined for every ordinal $\alpha$. The former are cylindric algebras of dimension $\alpha$ expanded with $\sf S4$ modalities indexed by $\alpha$. 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 varieties containing and including the variety of representable algebras of dimension $n$ are not atom canonical. We show that any class containing the class of completely representable algebras having a weak neat embedding property is not elementary. From these two results we draw the same conclusion on omitting types for finite variable fragments of predicate topologic and temporal logic. We show that the usual version of the omitting types theorem restricted to such fragments when the number of variables is $>2$ fails dramatically even if we considerably broaden the class of models permitted to omit a single non principal type in countable atomic theories, namely, the non-principal type consting of co atoms.Comment: arXiv admin note: substantial text overlap with arXiv:1308.6165, arXiv:1307.1016, arXiv:1309.0681, arXiv:1307.4298, arXiv:1401.1103, arXiv:1401.115

For finite n\geq 3, and k\geq 4, the variety SNr_n\CA_{n+k} is not atom canonical

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.Comment: arXiv admin note: text overlap with arXiv:1305.4570, arXiv:1302.1368, arXiv:1305.5269, arXiv:1304.078

Finite relation algebras and omitting types in modal fragments of first order logic

Let 2<n\leq l<m< \omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n--variable fragments of first order logic with respect to clique guarded semantics, and for its packed n--variable fragments. Both are modal fragments of L_n. As a sample, we show that if there exists a finite relation algebra with a so--called strong l--blur, and no m--dimensional relational basis, then there exists a countable, atomic and complete L_n theory T and type \Gamma, such that \Gamma is realizable in every so--called m--square model of T, but any witness isolating \Gamma cannot use less than $l$ variables. An $m$--square model M of T gives a form of clique guarded semantics, where the parameter m, measures how locally well behaved M is. Every ordinary model is k--square for any n<k<\omega, but the converse is not true. Any model M is \omega--square, and the two notions are equivalent if M is countable. Such relation algebras are shown to exist for certain values of l and m like for n\leq l<\omega and m=\omega, and for l=n and m\geq n+3. The case l=n and m=\omega gives that the omitting types theorem fails for L_n with respect to (usual) Tarskian semantics: There is an atomic countable L_n theory T for which the single non--principal type consisting of co--atoms cannot be omitted in any model M of T. For n<\omega, positive results on omitting types are obained for L_n by imposing extra conditions on the theories and/or the types omitted. Positive and negative results on omitting types are obtained for infinitary variants and extensions of L_{\omega, \omega}.Comment: arXiv admin note: text overlap with arXiv:1408.3282, arXiv:1502.0770

Amalgmation in Boolean algebras with operators

We study various forms of amalgamation for Boolean algebras with operations. We will also have the occasion to weaken the Boolean structure dealing with MV and BL algebras with operators.Comment: arXiv admin note: substantial text overlap with arXiv:1302.3043, arXiv:1303.7386, arXiv:1304.0612, arXiv:1304.114

Interpolation in many valued predicate logics using algebraic logic

Using polyadic MV algebras, we show that many predicate many valued logics have the interpolation property.Comment: 49 pages. arXiv admin note: text overlap with arXiv:1304.070

Cylindric polyadic algebras have the superamalgamation

We show that cylindric polyadic algebras introduced by Ferenczi has the superamalgmation property. We give two proofs. One is a Henkin construction, and the other is inspired by duality theory in modal logic between finite zig zag products of Kripke frames and their complex algebras.Comment: arXiv admin note: substantial text overlap with arXiv:1303.738

Neat atom structures

An atom structure is neat if there an algebra based on this atom structure in Nr_nCA_{\omega}. We show that this class is not elementaryComment: arXiv admin note: text overlap with arXiv:1302.136

Free algebras, amalgamation, and a theorem of Vaught for many valued logics

We investigate atomicity of free algebras and various forms of amalgamation for BL and MV algebras, and also Heyting algebras, though the latter algebras may not be linearly ordered, so strictly speaking their corresponding intuitionistic logic does not belong to many valued logic. Generalizing results of Comer proved in the classical first order case; working out a sheaf duality on the Zarski topology defined on the prime spectrum of such algebras, we infer several definability theorems, and obtain a representation theorem for theories as continuous sections of Sheaves. We also prove an omitting types theorem for fuzzy logic, and formulate and prove several of its consequences (in classical model theory) adapted to our case; that has to do with existence and uniqueness of prime and atomic models.Comment: arXiv admin note: substantial text overlap with arXiv:1304.0707, arXiv:1304.0612, arXiv:1304.0760, arXiv:1302.304