Generating Random Elements of Finite Distributive Lattices

This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling from the past'' to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, and alternating-sign matrices.Comment: 13 page

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page

Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions

The modular structure of an ontology: Atomic decomposition

Extracting a subset of a given ontology that captures all the ontology’s knowledge about a specified set of terms is a well-understood task. This task can be based, for instance, on locality-based modules. However, a single module does not allow us to understand neither topicality, connectedness, structure, or superfluous parts of an ontology, nor agreement between actual and intended modeling. The strong logical properties of locality-based modules suggest that the family of all such modules of an ontology can support comprehension of the ontology as a whole. However, extracting that family is not feasible, since the number of localitybased modules of an ontology can be exponential w.r.t. its size. In this paper we report on a new approach that enables us to efficiently extract a polynomial representation of the family of all locality-based modules of an ontology. We also describe the fundamental algorithm to pursue this task, and report on experiments carried out and results obtained.