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

On neat atom structures for cylindric like algebras

(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly 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=\alpha, \A\in \Nr_n\CA_{\omega} and for every algebra \B based on this atom structure there exists k\in \omega, k\geq 1, such that \B\in \Nr_n\CA_{n+k}. (2) Let k\leq \omega. Call an atom structure \alpha k complete, if there exists \A such that \At\A=\alpha and \A\in S_c\Nr_n\CA_{n+k}. (3) Let k\leq \omega. Call an atom structure \alpha$ k neat if there exists \A such that \At\A=\alpha, and \A\in \Nr_n\CA_{n+k}. (4) Let K\subseteq \CA_n, and \L be an extension of first order logic. We say that \K is well behaved w.r.t to \L, if for any \A\in \K, A atomic, and for any any atom structure \beta such that \At\A is elementary equivalent to \beta, for any \B, \At\B=\beta, then B\in K. We investigate the existence of such structures, and the interconnections. We also present several K's and L's as in the second definition. All our results extend to Pinter's algebras and polyadic algebras with and without equality.Comment: arXiv admin note: substantial text overlap with arXiv:1305.453

Neat embeddings as adjoint situations

We view the neat reduct operator as a functor that lessens dimensions from CA_{\alpha+\omega} to CA_{\alpha} for infinite ordinals \alpha. 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.Comment: arXiv admin note: substantial text overlap with arXiv:1303.738

Results on Polyadic Algebras

While every polyadic algebra (\PA) of dimension 2 is representable, we show that not every atomic polyadic algebra of dimension two is completely representable; though the class is elementary. Using higly involved constructions of Hirsch and Hodkinson we show that it is not elementary for higher dimensions a result that, to the best of our knowledge, though easily destilled from the literature, was never published. We give a uniform flexible way of constructing weak atom structures that are not strong, and we discuss the possibility of extending such result to infinite dimensions. Finally we show that for any finite $n>1$, there are two $n$ dimensional polyadic atom structures \At_1 and \At_2 that are $L_{\infty,\omega}$ equivalent, and there exist atomic \A,\B\in \PA_n, such that \At\A=\At_1 and \At\B= \At_2, \A\in \Nr_n\PA_{\omega} and \B\notin \Nr_n\PA_{n+1}. This can also be done for infinite dimensions (but we omit the proof)Comment: arXiv admin note: text overlap with arXiv:1304.114

Problems on neat embeddings solved by rainbow constructions and Monk algebras

This paper is a survey of recent results and methods in (Tarskian) algebraic logic. We focus on cylindric algebras. Fix 2<n<\omega. 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 compared to constructions using Monk-like algebras and cases where both constructions work are given. When splitting methods fail. rainbow constructions are used to show that diagonal free varieties of representable diagonal free algebras of finite dimension n, do no admit universal axiomatizations containing only finitely many variables. Notions of representability, like complete, weak and strong are lifted from atom structures to atomic algebras and investigated in terms of neat embedding properties. The classical results of Monk and Maddux on non-finite axiomatizability of the classes of representable relation and cylindric algebras of finite dimension n are reproved using also a blow up and blur construction. Applications to n-variable fragments of first order logic are given. The main results of the paper are summarized in tabular form at the end of the paper.Comment: arXiv admin note: substantial text overlap with arXiv:1408.328

Atomic polyadic algebras of infinite dimension are completely representable

We show that atomic polyadic algebras of infinite dimensions are completely representabl

Strongly representable atom structures and neat embeddings

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 representations. Finally, we show that for n>5, the problemn as to whether a finite algebra is in the class SNr_3CA_6 is undecidable. In contrast, we show that for a finite algebra of arbitary finite dimensions that embed into extra dimensions of a another finite algebra, then this algebra have a finite relativized representation. Finally we devise what we call neat games, for such a game if \pe\ has a \ws \ on an atomic algebra \A in certain atomic game and \pa has a \ws in another atomic game, then such algebras are elementary equivalent to neat reducts, but do not have relativized (local) complete represenations. From such results, we infer that the omitting types theorem for finite variable fragments fails even if we consider clique guarded semantics. The size of cliques are determined by the number of pebbles used by \pa\.Comment: arXiv admin note: text overlap with arXiv:1302.1368, arXiv:1304.1149, arXiv:1305.4570, arXiv:1307.101

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

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

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