Conference on the Frames and Spaces of Ordered Algebraic Structures

University Of Florida, Gainesville FL

Investigators

Abstract

This conference is held at the University of Florida, from March 9th through the 11th, 2006. It is referred to as Ord06/UF This conference is the ninth in a series initiated in 1998; these conferences 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 establishes a 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.