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We study extensions of standard description logics to the framework of polyadic modal logic. We promote a natural approach to such logics via general relation algebras that can be used to define operations on relations of all arities. As a…
In this paper we provide explicit dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and…
We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by (non)-abelian groups…
Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to…
This paper presents a many-sorted polyadic modal logic that generalizes some of the existing approaches. The algebraic semantics has led us to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue…
We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections,…
We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent…
The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a…
We extend Wallman's classic duality from lattice bases to semilattice subbases and from compact to locally closed compact spaces. Moreover, we make this duality functorial via appropriate relational morphisms.
The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although…
We define certain class of correspondences of polarized representations of $C^*$-algebras. Our correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms joining two manifolds. In this setup we define the…
Two main topics are considered: The characterisation of finite homomorphism dualities for relational structures, and the splitting property of maximal antichains in the homomorphism order.
We show for the moduli space of rank-2 coherent sheaves on an algebraic surface that there exists a 'dual' moduli space. This dual space allows a construction of the first one without using the GIT construction. Furthermore, we obtain a…
We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these…
This chapter presents a state-of-the-art survey of relationships, traditionally referred to as `bridges', between interpolation properties for propositional logics -- including superintuitionistic, modal, and substructural logics -- and…
We survey systematic approaches to basis-restricted fragments of propositional logic and modal logics, with an emphasis on how expressive power and computational complexity depend on the allowed operators. The propositional case is…
This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of…
We develop a framework for a duality theory for general multilinear operators which extends that for transversal multilinear operators which has been established in arXiv:1809.02449. We apply it to the setting of joints and multijoints, and…
This work is divided between two main areas: in the theory of multialgebras, we focus mostly on a new definition of what a freely generated object should be in their category, and on how this category is equivalent to another with partially…
In this paper, we generalize modal $\mu$-calculus to the non-distributive (lattice-based) modal $\mu$-calculus and formalize some scenarios regarding categorization using it. We also provide a game semantics for the developed logic. The…