We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x ∈ D | a ≤ b ∨ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x ≤ y ∨(x-y),(x-y)∧(y-x) = 0, and x-z ≤ (x-y)∨(y-z). In particular, D is not a homomorphic image of the ...

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian ℓ-group with unit (resp., ...

It is shown that by introducing a gauge group parameter in the path integral formulation of an abelian gauge theory, the preservation of gauge invariance of the effective action is allowed, and at the same time, can be made to preserve current conservation. The procedure introduced here is shown to yield a theory in which there is no anomaly.

We consider a group G that does not have the independence property and study the definability of certain subgroups of G using parameters from a fixed elementary extention G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. W...