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

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

Algebraisable versions of predicate topological logic

Motivated by questions like: which spatial structures may be characterized by means of modal logic, what is the logic of space, how to encode in modal logic different geometric relations, topological logic provides a framework for studying the confluence of the topological semantics for $\sf S4$ modalities, based on topological spaces rather than Kripke frames. Following research initiated by Sgro, and further pursued algebraically by Georgescu, we prove an interpolation theorem and an omitting types theorem for various extensions of predicate topological logic and Chang's modal logic. Our proof is algebraic addressing expansions of cylindric algebras using interior operators and boxes, respectively. Then we proceed like is done in abstract algebraic logic by studing algebraisable extensions of both logics; obtaining a plethora of results on the amalgamation property for various subclasses of their algebraic counterparts, which are varieties. Notions like atom-canonicity and complete representations are approached for finite dimensional topological cylindric algebras. The logical consequences of our algebraic results are carefully worked out for infinitary extensions of Chang's predicate modal logic and finite versions thereof, by restricting to $n$ variables, $n$ finite, viewed as a propositional multi-dimensional modal logic, and $n$ products of bimodal whose frames are of the form $(U, U\times U, R)$ where $R$ is a pre-order, endowed with diagonal constants