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The conference on Ordered Algebraic Structures held in Curat;ao,
from the 26th of June through the 30th of June, 1995, at the Avila
Beach Hotel, marked the eighth year of ac tivities by the Caribbean
Mathematics Foundation (abbr. CMF), which was the principal sponsor
of this conference. CMF was inaugurated in 1988 with a conference
on Ordered Algebraic Structures. During the years between these two
conferences the field has changed sufficiently, both from my point
of view and, I believe, that of my co-organizer, W. Charles
Holland, to make one wonder about the label "Ordered Algebraic
Structures" itself. We recognized this from the start, and right
away this conference carried a subtitle, or, if one prefers, an
agenda: we concentrated on the one hand, on traditional themes in
the theory of ordered groups, including model-theoretic aspects,
and, on the other hand, on matters in which topology (more
precisely C(X)-style topology) and category theory would play a
prominent role. Plainly, ordered algebra has many faces, and it is
becoming increas ingly difficult to organize an intimate
conference, such as the ones encouraged in the series sponsored by
CMF, in this area on a broad set of themes. These proceedings
reflect, accurately we think, the spirit of the conferees, but it
is not a faithful record of the papers presented at the
conference."
This volume contains a selection of papers presented at the 1991
Conrad Conference, held in Gainesville, Florida, USA, in December,
1991. Together, these give an overview of some recent advances in
the area of ordered algebraic structures. The first part of the
book is devoted to ordered permutation groups and universal, as
well as model-theoretic, aspects. The second part deals with
material variously connected to general topology and functional
analysis. Collectively, the contents of the book demonstrate the
wide applicability of order-theoretic methods, and how ordered
algebraic structures have connections with many research
disciplines. For researchers and graduate students whose work
involves ordered algebraic structures.
The subjects of ordered groups and of infinite permutation groups
have long en joyed a symbiotic relationship. Although the two
subjects come from very different sources, they have in certain
ways come together, and each has derived considerable benefit from
the other. My own personal contact with this interaction began in
1961. I had done Ph. D. work on sequence convergence in totally
ordered groups under the direction of Paul Conrad. In the process,
I had encountered "pseudo-convergent" sequences in an ordered group
G, which are like Cauchy sequences, except that the differences be
tween terms of large index approach not 0 but a convex subgroup G
of G. If G is normal, then such sequences are conveniently
described as Cauchy sequences in the quotient ordered group GIG. If
G is not normal, of course GIG has no group structure, though it is
still a totally ordered set. The best that can be said is that the
elements of G permute GIG in an order-preserving fashion. In
independent investigations around that time, both P. Conrad and P.
Cohn had showed that a group admits a total right ordering if and
only if the group is a group of automor phisms of a totally ordered
set. (In a right ordered group, the order is required to be
preserved by all right translations, unlike a (two-sided) ordered
group, where both right and left translations must preserve the
order."
This volume contains a selection of papers presented at the 1991
Conrad Conference, held in Gainesville, Florida, USA, in December,
1991. Together, these give an overview of some recent advances in
the area of ordered algebraic structures. The first part of the
book is devoted to ordered permutation groups and universal, as
well as model-theoretic, aspects. The second part deals with
material variously connected to general topology and functional
analysis. Collectively, the contents of the book demonstrate the
wide applicability of order-theoretic methods, and how ordered
algebraic structures have connections with many research
disciplines. For researchers and graduate students whose work
involves ordered algebraic structures.
The subjects of ordered groups and of infinite permutation groups
have long en joyed a symbiotic relationship. Although the two
subjects come from very different sources, they have in certain
ways come together, and each has derived considerable benefit from
the other. My own personal contact with this interaction began in
1961. I had done Ph. D. work on sequence convergence in totally
ordered groups under the direction of Paul Conrad. In the process,
I had encountered "pseudo-convergent" sequences in an ordered group
G, which are like Cauchy sequences, except that the differences be
tween terms of large index approach not 0 but a convex subgroup G
of G. If G is normal, then such sequences are conveniently
described as Cauchy sequences in the quotient ordered group GIG. If
G is not normal, of course GIG has no group structure, though it is
still a totally ordered set. The best that can be said is that the
elements of G permute GIG in an order-preserving fashion. In
independent investigations around that time, both P. Conrad and P.
Cohn had showed that a group admits a total right ordering if and
only if the group is a group of automor phisms of a totally ordered
set. (In a right ordered group, the order is required to be
preserved by all right translations, unlike a (two-sided) ordered
group, where both right and left translations must preserve the
order."
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