Positivity 8: 305–313, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Regular Ordering and Existence ofMinimum
Points in Uniform Spaces and Topological Groups
A. B. NÉMETH
Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 3400 Cluj-Napoca,
Romania. E-mail: email@example.com
(Received 10 October 2002; accepted 24 July 2003)
Abstract. It is deﬁned the locally order convexity ofan ordered uniform space and an ordered
topological group and is investigated its relation to the existence ofminimum points ofdenumerable,
complete lower bounded sets.
AMS Subject Classiﬁcations 1991: Primary: 54F05; Secondary: 58E17, 58E30, 06F20
Key words: Ordered uniform spaces, Ordered topological groups
Ordered uniform spaces appear implicitly in variational problems [1, 9]. Appar-
ently an attempt to a direct approach appears ﬁrst in . This last paper focuses
on applications. In the present note we consider some locally order convexity type
notions (frequently used in ordered topological vector space theory) for the ordered
uniform spaces and their relation to minimization problems in ordered uniform
spaces and ordered topological groups.
Let us consider a nonempty set E. An arbitrary subset ⊂E ×E is called a
relation (on E). We shall write xy if xy∈ and x
y if yx ∈.
called the opposite ofthe relation .IfF is a subset ofE, then ∩F ×Fis called
the relation induced by in F . Two elements x and y in E are called -comparable
if xy or yx. A subset A ∈E is called an -chain ifany two elements ofits are
If and are relations, then the relation = is deﬁned by xy ifand
only ifthere exists some element z with the property that xz and zy.
The relation is said reﬂexive if = xxx ∈E⊂ . It is called
transitive if ⊂. It is called antisymmetrical,if∩