Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure. 1.

Monotonic modal logics form a generalization of normal modal logics...

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion. 1

Algebraic work [9] shows that the deep theory of possible world semantics is available in the more general setting of substructural logics, at least in an algebraic guise. The question is whether it is also available in a relational form. This article seeks to set the stage for answering this question. Guided by the algebraic theory, but purely relationally we introduce a new type of frames. These structures generalize Kripke structures but are two-sorted, containing both worlds and co-worlds. These latter points may be viewed as modelling irreducible increases in information where worlds model irreducible decreases in information. Based on these structures, a purely model theoretic and uniform account of completeness for the implication-fusion fragment of various substructural logics is given. Completeness is obtained via a generalization of the standard canonical model construction in combination with correspondence results. 1

Abstract. From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that Θ = {(U, V) | U ∩ A = V ∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.

The canonical extension of a lattice is in an essential way a two-sided completion. Domain theory, on the contrary, is primarily concerned with one-sided completeness. In this paper, we show two things. Firstly, that the canonical extension of a lattice can be given an asymmetric description in two stages: a free co-directed meet completion, followed by a completion by selected directed joins. Secondly, we show that the general techniques for dcpo presentations of dcpo algebras used in the second stage of the construction immediately give us the well-known canonicity result for bounded lattices with operators. Key words: dcpo presentation, dcpo algebra, lattice theory, canonical extension, canonicity

Abstract. We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the one given e.g. in [3], but it agrees with it whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense. 1.

Canonical extensions of double quasioperator algebras: an algebraic perspective on duality for certain algebras with binary operations

We develop an algebraic modal logic that combines epistemic modalities with dynamic modalities with a view to modelling information acquisition (learning) by automated agents in a changing world. Unlike most treatments of dynamic epistemic logic, we have transitions that “change the state ” of the underlying system and not just the state of knowledge of the agents. The key novel feature that emerges is the need to have a way of “inverting transitions” and distinguishing between transitions that “really happen ” and transitions that are possible. Our approach is algebraic, rather than being based on a Kripke-style semantics. The semantics are given in terms of quantales. We study a class of quantales with the appropriate inverse operations and prove soundness and completeness theorems. We illustrate the ideas with a simple game as well as a toy robot-navigation problem. The examples illustrate how an agent discovers information by taking actions. 1

we focus on the canonical extension of partially ordered sets, which was defined by algebraic means by Dunn, Gehrke and Palmigiano [9]. We show that it can be obtained alternatively via a generalization of Urquhart and Hartung’s maximal filter-ideal pair construction ([43], [27]). We further give a first-order dual characterization of perfect lattice hemiand homomorphisms, in the spirit of, but different from Gehrke [16], and make category-theoretic observations regarding the canonical extension. The second part of the thesis concerns the algebraic canonicity proof of the Sahlqvist fragment for distributive modal logic by Gehrke, Nagahashi and Venema [22]. We pay particular attention to the additional operation n, which is crucial to that proof, and show that the proof can not be straightforwardly translated to an algebraic canonicity proof of the inductive fragment for distributive modal logic [7]. We extract requirements on a new version of the operation n, which would yield a proof of the canonicity of the inductive fragment, and finish by starting to explore two new perspectives on the magical nature of the operation n. Contact information