Varieties of Lattice-Ordered Groups.
Almost finite-valued -groups
Dimension in algebraic frames
In an algebraic frame the dimension, , is defined, as in classical ideal theory, to be the maximum of the lengths of chains of primes , if such a maximum exists, and otherwise. A notion of “dominance” is then defined among the compact elements of , which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of , including the frames and of -elements and -elements, respectively. The more concrete illustrations...
Free products of abelian -groups
Free products in varieties of lattice-ordered groups
Torsion theory for lattice-ordered groups
Prime selectors in lattice-ordered groups
Archimedean frames, revisited
This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called and is . Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an element arises naturally, and the join of suitably chosen infinitesimals...
Torsion theory for lattice-ordered groups Part II: Homogeneous -groups
Pairwise splitting lattice-ordered groups
Locally conditioned radical classes of lattice-ordered groups
Functorial rings of quotients 2: the ring of hyper-fractions.
Functorial approximation to the lateral completion in archimedean lattice-ordered groups with weak unit
Dimension in algebraic frames, II: Applications to frames of ideals in
This paper continues the investigation into Krull-style dimensions in algebraic frames. Let be an algebraic frame. is the supremum of the lengths of sequences of (proper) prime elements of . Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of in terms of the dimensions of certain boundary quotients of . This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...
-points vs -points and -points
In a Tychonoff space , the point is called a -point if every real-valued continuous function on can be extended continuously to . Every point in an extremally disconnected space is a -point. A classic example is the space consisting of the countable ordinals together with . The point is known to be a -point as well as a -point. We supply a characterization of -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a -point....
Locally finite conditions on lattice-ordered groups
On adjoining units to hyper-archimedean -groups
Algebras and spaces of dense constancies
A DC-space (or space of dense constancies) is a Tychonoff space such that for each there is a family of open sets , the union of which is dense in , such that , restricted to each , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean -algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions...
Spaces in which all prime -ideals of are minimal or maximal
Quasi -spaces are defined to be those Tychonoff spaces such that each prime -ideal of is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of -spaces. The compact quasi -spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi -spaces is given. If is a cozero-complemented space and every nowhere dense zeroset...
A hit-and-miss topology for , Cₙ(X) and Fₙ(X)
A hit-and-miss topology () is defined for the hyperspaces , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between and the Vietoris topology and we find conditions on X for which these topologies are equivalent.
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