We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...

We prove that for every distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) ≤ a ∨ b, then there are a pos...

We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...

The universal abelian, band, and semilattice compactifications of a semitopological semigroup are characterized in terms of three function algebras. Some relationships among these function algebras and some well-known ones, from the universal compactification point of view, are also discussed.

Requirements engineering has, at least, three dimensions: (1) a logistical dimension dealing with requirements management, requirement databases and tools, (2) a methodological aspect dealing with issues such as “how to elicit requirements?”, “how to analyze and how to validate requirements?” and (3) an epistemological dimension which deals with questions such as “what is a well-formed requirem...

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly ^-reflexive inverse subsemigroups need not be strongly £-reflexive. If, however, the union is saturated with respect to the Green's relation <3), and in particular if the union is a disjoint one, then 5 is indeed strongly £-reflexive. This is established by showing that fy -saturated inverse sub...

The lattice of flats of a matroid or combinatorial geometry can be regarded as a sublattice (with rank function) of the lattice of subsets of a set, having the property that, given an element of rank r of the lattice and a point outside it, a unique element of rank r+1 covers both. This paper develops a theory of permutation geometries, which are similarly defined objects in the semilattice of ...

Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a 7→ a−1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have e...