arXiv subjects
Publications and source records attributed to Tarek Sayed Ahmed.
We prove the celebrated representation theorem of Andreka-Resek-Thompson, together with its polyadic analogue by Ferenczi, using games as introduced in algebraic logic by Hirsch and Hodkinson. We also show that atomic algebras are completely representable and that all such varieties of relativized set algebras have the strong amalgmation property.
Usually when we have polyadic-like algebras, meaning that we have infinitary substitutions (that is substitutions moving infinitely many points) in the similarity type, then we get the superamalgamation property especially if this class of algebras happen to be a variety. This for example happens for (full) polyadic algebras, full Heyting algebras and reducts of those using only finitely many infinitary substitutions, like Sains Boolean and Heyting algebras. (The last is studied by Sayed Ahmed) . In cylindric-like algebras (like quasi-polyadic algebras) when we do not have infinitary substitutions, we do not get even the amalgamation property . In this paper, we give an example of a polyadic like variety (we have infinitary substitutions, in fact infinitely many of them) for which the amalgamation property fails.
Extending a deep result of Andreka and Nemeti, we show that unlike the dimension complemented case, there are weak set quasi-polyadic simple algebras of dimension >1, that are finitely genertaed with more than one element, but cannot be generated with a single element. We give a contrasting result for polyadic agebras of infinite dimension.
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.
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 ϕbe a first order formula and M be a countable model. ϕ^M denotes the set of all assignments that satisfy ϕin M. Let M, N be countable models. A formula ϕdistinguishes these models if |ϕ^M|\neq |ϕ^N|. We show that the number of distinguishable countable models for a complete countable first order theory, satisfies the Glimm Effross dichotomy, hence also Vaught's conjecture. The proof in the presence of equality is fairly easy, though we need infinite conjunctions to implement it and somewhat heavy theorems from descriptive set theory. However, the case without equality seems to be much more difficult (because we cannot count without equality) and in this case algebraic logic comes to our rescue via the representation theory of locally finite quasi polyadic algebras. We show that the equivalence relation induced by such (indistinguishable) models is a Borel subset of the product of the Stone Polish space (consisting of ultarfilters corresponding to models) with itself. We also formuate and prove a Vaught's theorem for certian infinitary extensions of first order logic via the representation theory of dimension complemented algebras. In all case we also count the number of non isomorphic models omitting < covK many non isolated types (the latter is the least cardinal such that the Baire Category theorem fails and also the largest cardinal such tha Martin's axiom restrcited to countable Boolean algebra hold.) In such cases, for locally finite algebras and dimension complemented ones we obtain the same number of non-ismorphic distinguishable models.
We prove several representation theorems for infinitary predicate modal logic
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.
Several open problems in algebraic logic are solved.
We prove completeness, interpolation and omitting types for certain predicate topological logics that properly extend the first order case. We aslo count the non isomorphic topological models of a countable theory
Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor. We give applications to many-valued logics logics and various modifications of first order logic and multi-modal logic, set in an algebraic framework.
We answer an implicit question of Ian Hodkinson's. We show that atomic Pinters algebras may not be completely representable, however the class of completely representable Pinters algebras is elementary and finitely axiomatizable. We obtain analagous results for infinite dimensions (replacing finite axiomatizability by finite schema axiomatizability). We show that the class of subdirect products of set algebras is a canonical variety that is locally finite only for finite dimensions, and has the superamalgamation property; the latter for all dimensions. However, the algebras we deal with are expansions of Pinter algebras with substitutions corresponding to tranpositions. It is true that this makes the a lot of the problems addressed harder, but this is an acet, not a liability. Futhermore, the results for Pinter's algebras readily follow by just discarding the substitution operations corresponding to transpostions. Finally, we show that the multi-dimensional modal logic corresponding to finite dimensional algebras have an $NP$-complete satisfiability problem.
We show that several reducts of Heyting polyadic algebras of infinite dimension, with and without equality enjoy various amalgamation properties. In the equality free case we obtain superamalgamation, but when we have equality we obtain a weaker interpolation property, for the corresponding infinitary intuitionistic logic with equality.
We characterize completey (give a necessary and suffcient condition using special neat embeddings)for a relation algebra to belong to the amalgamation, strong amalgamation, and superamalgamation base of the class of representable algebras. We do the same for cylindric and polyadic algebras for all dimensions >1, infinite included. Finally, we show that we can expand our algebras by finitely many natural operations, that force the newly expanded algebras to have various forms of amalgamation properties, like quasi-projections and directed cylindrifiers.
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.
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.
Using polyadic MV algebras, we show that many predicate many valued logics have the interpolation property.
We show that quasi-projective relation algebras and directed cylindric algebras are equivalent categorialy. We work out a Godels second incompleteness theorem for finite varibale fragments of first order logic. We show that distinct set theories (like one with CH, and another with its negation) give rise to equationally distinct simple directed cylindric algebras. This correspondance was worked out for quasi projective relation algebras (with a distinguished element corresponding to membership relation). The idea of the proof is that the Sagi representation of directed cylindric algebras preserve well-foundnes. Finally, using that the functor defined preserves order, we show that the class of directed cylindric algebras have the superamalgmation property.