CoproductMV-Algebras, Nonstandard Reals, and Riesz Spaces

Abstract Up to categorical equivalence,MV-algebras are unit intervals of abelian lattice-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 anyMV-algebraAas particular cases of the general algebraic notions. The radical Rad Ais the intersection of all maximal ideals ofA. AnMV-algebraAis said to be local iff it has a unique maximal ideal. Then, by Hoelder's theorem, the quotientA/Rad Ais isomorphic to a subalgebra of the real unit interval [0, 1]. Using nonstandard real numbers we give a concrete representation of those totally orderedMV-algebrasAwhich are isomorphic to the coproduct ofA/Rad Aand 〈Rad A〉, the latter denoting the subalgebra ofAgenerated by its radical. As an application, using several categorical equivalences we describe theMV-algebraic counterparts of Riesz spaces, also known as vector lattices.