Bgsu Mini-course on Baker-beynon Duality

This document contains a rough summary of the main theorems discussed during the mini-course on Baker-Beynon Duality for unital latticeordered Abelian groups and MV-algebras held at Bowling Green State University from 8.VI.2009 through 12.VI.2009, with references to the original sources. Bibliographic Note. The standard reference on MV-algebras is [10]. It develops all of the basic theory in careful detail. It does not cover the more advanced results, nor the more recent ones. In [10], Baker-Beynon duality for MV-algebras is not really dealt with, although much useful material needed for its development is to be found there. Regrettably, at the time of writing I know of no single source that adequately covers Baker-Beynon duality for MV-algebras or for unital latticeordered Abelian groups. Because MV-algebras were introduced as the algebraic counterpart of Lukasiewicz infinite-valued propositional logic, there are some books in mathematical logic that deal with MV-algebras; in this context, [16] is a standard reference. There are several standard references for partially ordered and lattice-ordered groups. For those interested in doing research in the subject, it is definitely worthwhile to get acquainted with as many of them as possible. Let me mention at least [7], still a source of inspiration to this day; [13], with a nice early treatment of general partially ordered algebraic systems; [6], with the most complete treatment of the prime spectrum as a topological space to be found to date in handbooks on lattice-ordered groups; [1], which includes a treatment of groups of divisibility, along with a useful, annotated list of examples of lattice-ordered groups with special properties; [11], with a coverage that is both broad and deep, and with an authoritative treatment of varieties; [20], with several things not to be found elsewhere, such as the description of all total orders on free Abelian groups of finite rank in §6.3; [14], which includes both an account of order-preserving permutations by a leading expert, and a useful partial coverage of Baker-Beynon duality for the non-unital case; and finally [15], an important book that deals with the connection between dimension groups, a class of partially-ordered Abelian groups larger than the lattice-ordered ones, and C∗-algebras, certain rings with additional structure that are investigated in functional analysis and mathematical physics. Day 1 – Monday, 8.VI.2009. Preliminaries. We introduced MV-algebras and lattice-ordered Abelian groups with a distinguished (strong order) unit, unital Abelian `-groups for short. As important examples, we introduced (R, 1), the totally ordered Abelian group of real numbers (under addition) with its natural order and Date: August 31, 2009.