Every endofunctor B on the category Set can be lifted to a fibred functor on the category (fibred over Set) of equivalence relations and relation-preserving functions. In this paper, the least (fibre-wise) of such liftings, L(B), is characterized for essentially any B. The lifting has all the useful properties of the relation lifting due to Jacobs, without the usual assumption of weak pullback ...

We construct a ‘weak’ version EMw(K ) of Lack & Street’s 2-category of monads in a 2-category K , by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EMw(K ) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of ‘wea...

We construct Boolean-valued term models and describe their universal properties. The underlying set of these models consists of the terms of the underlying language, while formulae are interpreted in the Lindenbaum-Tarski algebra of open formulae. Subspaces of the Stone space of the Lindenbaum-Tarski algebra of open formulae are shown to be homeomorphic to spaces of enumerated models models equ...

We study the concept of finitary formal topology, a point-free version of a topological space with a basis of compact open subsets. The notion of finitary formal topology is defined from the perspective of the Basic Picture (introduced by the second author) and thus it is endowed with a binary positivity relation. As an application, we prove a constructive version of Stone’s representation theo...

Some aspects of the theory of Boolean algebras and distributive lattices -in particular, the Stone Representation Theorems and the properties of filters and ideals -are analyzed in a constructive setting.

Let $H$ be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:1}$ appeared in Kashina's work \cite{K00}. We obtain all simple Yetter-Drinfeld modules over then determine finite-dimensional Nichols algebras satisfying $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a object $_H^H\math...

We extend the Sherali-Adams, Lovász-Schrijver, Balas-Ceria-Cornuéjols and Lasserre lift-and-project methods for 0-1 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a set-covering problem at least as strong as that given by the set of all valid inequalities with small coefficients, and, more generally, all valid ine...

Starting with the motivating example of Stone’s representation theorem that allows one to represent Boolean algebras as subalgebras of the poweralgebra of a sufficiently large set, we ask the question of whether it is possible to generalize this to a relationship between lattice theory and topology. This can be done by considering special lattices called locales, which are, in a sense, a suitab...