Construction of lattice orders on the semigroup ring of a positive rooted monoid

✍ Scribed by Jingjing Ma; Stuart A. Steinberg

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 247 KB
• Volume: 260
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00053-x

✦ Synopsis

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.