Contexts where entity descriptions belong to a meet-semilattice are considered. When the entity set is finite, we show that nonempty extensions of concepts assigned to such contexts coincide, casewise, with strong or weak clusters associated with some pairwise or multiway dissimilarity measure. Moreover, by duality principle, a similar result holds when entity descriptions belong to a join-semi...

Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice the Alexander dual is computed.

We begin by recalling the general theory of adjoints on finite semilattices. A finite join semilattice with 0 is a lattice, with the naturally induced meet operation. Thus a finite lattice S can be regarded as a semilattice in two ways, either S = 〈S,+, 0〉 or S = 〈S,∧, 1〉. Given a (+, 0)-homomorphism g : S → T , define the adjoint h : T → S by h(t) = ∑ {s ∈ S : gs ≤ t} so that gs ≤ t iff s ≤ ht...

I define the concepts of multifuncoid (and completary multifuncoid) and upgrading. Then I conjecture that upgrading of certain multifuncoids are multifuncoids (and that upgrading certain completary multifuncoids are completary multifuncoids). I have proved the conjectures for n 6 2. This short article is the first my public writing where I introduce the concept of multidimensional funcoid which...

The aim of this work is the formalization of Chapter 0 Section 4 of [11]. In this paper the definition of meet-continuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved. Let X, Y be non empty sets, let f be a function from X into Y , and let Z be a non empty subset of X. One can verify that f • Z is non empty. Let us note that every non empty relational structure which is reflexi...

For k randomly chosen subsets of [n] = {1,2, ,n} we consider the probability that the Boolean algebra, distributive lattice, and meet semilattice which they generate are respectively free, or all of 2*. In each case we describe a threshold function for the occurrence of these events. The threshold functions for freeness are close to their theoretical maximum values. 1 §

This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297–321. The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Loc,Set] to the double exponential SS X where S is the Sierpiński locale. Further PU (X) and PL(X) are shown to b...

Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class S∧ of (unital) meet semilattices. Any S ∈ S∧ embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt(S), is a compact and ∨∧ -dense exte...

partial frames provide a rich context in which to do pointfree structured and unstructured topology. a small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. these axioms are sufficiently general to include as examples bounded distributive lattices, $sigma$-frames, $ka...

We consider exponentiable objects in lax slices of Top with respect to the specialization order (and its opposite) on a base space B. We begin by showing that the lax slice over B has binary products which are preserved by the forgetful functor to Top if and only if B is a meet (respective, join) semilattice in Top, and go on to characterize exponentiability over a complete Alexandrov space B.