Conference on the Frames and Spaces of Ordered Algebraic Structures

本次会议于2006年3月9日至11日在佛罗里达大学举行。该会议是1998年发起的系列会议中的第九次会议；这些会议涉及到有序代数结构的某些方面。Ord06/UF专注于框架理论在主题中的作用。Ord06/UF，像之前的系列一样，汇集了来自世界各地的25到30名数学家，包括研究生和那些最近获得博士学位的人。计划出版第06/UF号文件。只要有对数学家的学术训练，研究和教学就一直试图将抽象作为理解具体和特殊的一种手段。因此，数字的算术性质被抽象为群和环的理论，这些代数结构被发明出来“解释”为什么实数，例如，它们的行为方式。另一派研究人员利用空间中数字或点之间距离的几何概念来发展拓扑空间的抽象概念，这种概念建立了一种不需要测量距离的接近感。在这两种概括中，目的始终是通过抽象的行为来捕捉更大的现象目录，而不是创造一个促使抽象的实例的深奥复制品。在这种精神下，有序代数结构主义者把实数的自然序，连同它的算术和拓扑方面，发展了格序群和环。Ord06/UF的贡献者，大体上都有一个共同的兴趣，那就是格序群如何最好地用函数来描述，这些函数的大部分值都是实数，并且这种构造的特征被传递到函数的代数中。在这种性质跟踪中，拓扑空间及其对框架的推广起着越来越重要的作用。特别是代数框架的概念具有双重身份，使人们能够同时捕获拓扑和算术性质，而且，一组性质如何影响另一组。

This conference is held at the University of Florida, from March 9th through the 11th, 2006. It is referred to as Ord06/UFThis conference is the ninth in a series initiated in 1998; theseconferences have involved some aspect of Ordered Algebraic Structures. Ord06/UF concentrates on the role of frame theory in the subject. Ord06/UF, like its predecessors in the series, brings together 25 to 30 mathematicians from various parts of the globe, including graduate students and those who have recently received their PhD degrees. A published proceedings of Ord06/UF is planned. For as long as there has been academic training of mathematicians, research and instruction have sought to abstract as a means towards understanding of the concrete and the particular. Thus, the arithmetical properties of numbers became abstracted to the theory of groups and rings, these being algebraic constructs invented to ``explain'' why the real numbers, for example, behave the way they do. Another school of researchers seized upon the geometric notion of distance between numbers or points in space to develop the abstract concept of a topological space, which establishesa sense of proximity without measurement of distance. In both of these generalizations the aim has consistently been to capture a greater catalogue of phenomena by the act of abstraction, and not to create an esoteric replica of the instance that prompted the abstraction. In this spirit the Ordered Algebraic Structurist has taken the natural ordering of real numbers, together with its arithmetical and topological aspects, and developed the lattice-ordered group and ring. The contributors to Ord06/UF, broadly, have in common an interest in how a lattice-ordered group may best be described by functions which take on most of their values in the real numbers, and in which features of the construct are carried over to the algebra of functions. In this tracking of properties, topological spaces and their generalization to frames have increasingly played an important role. The notion of an algebraic frame, in particular, has a dual identity, enabling one to simultaneously capture topological and arithmetical properties, and, moreover, how one set of properties influences the other.