Functional representation of substitution algebras

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.Comment: 8 page

The representations of polyadic-like equality algebras

It is stated that Boolean set algebras with unit V, where V is a union of Cartesian products, are axiomatizable. The axiomatization coincides with that of cylindric polyadic equality algebras (class CPE). This is an algebraic representation theorem for the class CPE by relativized polyadic set algebras in the class Gp. Similar representation theorems are claimed for the classes strong cylindric polyadic equality algebras (CPES) and cylindric m-quasi polyadic equality algebras (mCPE). These are polyadic-like equality algebras with infinite substitution operators and single cylindrifications. They can be regarded also as infinite transformation systems equipped with diagonals and cylindrifications. No representation theorem or neat embedding theorem has proven for this class of algebras yet, except for the locally finite case. The theorems occuring in the paper answer some unsolved problems.Comment: 5 page

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

The class of infinite dimensional quasipolaydic equality algebras is not finitely axiomatizable over its diagonal free reducts

We show that the class of infinite dimensional quasipolaydic equality algebras is not finitely axiomatizable over its diagonal free reduct

Representation theorems in modal logic using algebraic logic

We prove several representation theorems for infinitary predicate modal logi

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

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

What is the spirit of the cylindric paradigm, as opposed to that of the polyadic one?

We give a categorial definition separating cylindric-like algebras from polyadic-like ones. Viewing the neat reduct operator as a functor, we show that it does not have a right adjoint in the former case, but it is strongly invertible in the second case. Several new results on amalgamation, and non finite axiomatizability are presented for both paradigms. A hitherto categorial equivalence is also given between relation algebras with quasi-projections and Nemeti's directed cylindric algebras for any dimension.Comment: 88 pages. arXiv admin note: text overlap with arXiv:1302.036

On the multi dimensional modal logic of substitutions

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing (varieties generated by) complex algebras of Kripke semantics for such logic. Those algebras, whose elements are sets of states are common reducts of cylindric and polyadic algebra

Amalgamation, interpolation and congruence extension properties in topological cylindric algebras

Topological cylindric algebras of dimension \alpha, \alpha any ordinal are cylindric algebras with dimension \alpha expanded with \alpha S4 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 \alpha>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 axiomatizations and canonicity) and others, transfer to the topological addition, by implementing a very simple procedure of `discretely topologizing a cylindric algebra' Given a cylindric algebra of dimension \alpha, one adds \alpha many interior identity operations, the latter algebra is representable as a topological cylindric algebra if and only if the former is; the representation induced by the discrete topology. In this paper we investigate amalgamation properties for various classes of topological cylindric algebras of all dimensions. We recover, in the topological context, all of the results proved by Andreka, Comer Madarasz, Nemeti, Pigozzi, Sain, Sayed Ahmed, Sagi, Shelah, Simon, and others for cylindric algebras and much more