CoproductMV-Algebras, Nonstandard Reals, and Riesz Spaces

✍ Scribed by A. Di Nola; A. Lettieri

• Book ID: 102571399
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 214 KB
• Volume: 185
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0342

✦ Synopsis

Up to categorical equivalence, MV-algebras are unit intervals of abelian lattice-Ž . ordered groups for short, l-groups with strong unit. While the property of being a strong unit is not even definable in first-order logic, MV-algebras are definable by a few simple equations. Accordingly, such notions as ideals and coproducts are definable for any MV-algebra A as particular cases of the general algebraic notions. The radical Rad A is the intersection of all maximal ideals of A. An MV-algebra A is said to be local iff it has a unique maximal ideal. Then, by Hoelder's theorem, the quotient ArRad A is isomorphic to a subalgebra of the w x real unit interval 0, 1 . Using nonstandard real numbers we give a concrete representation of those totally ordered MV-algebras A which are isomorphic to ² : the coproduct of ArRad A and Rad A , the latter denoting the subalgebra of A generated by its radical. As an application, using several categorical equivalences we describe the MV-algebraic counterparts of Riesz spaces, also known as vector lattices.