Two themes that occur in this chapter concern the attempt to describe all lattice orders of a particular type on some familiar rings and the attempt to recognize some familiar lattice-ordered rings. Thus, a characterization is given of the real group ℓ-algebra of a locally finite group with the coordinatewise partial order and of the canonically ordered ℓ-group ring of a finite cyclic group with coefficients in a matrix ring over a totally ordered field. All of the lattice orders of a matrix algebra over a totally ordered subfield of the reals and of a 2×2 matrix algebra over any totally ordered field are determined. Also, for a polynomial ring in one variable over an ℓ-simple totally ordered Öre domain all nontotal lattice orders which extend the order of the coefficient ring and in which all squares are positive are described. This description is a specialization of a description of some lattice orders on a semigroup ring over a mopops.
KeywordsPartial Order Total Order Matrix Algebra Division Ring Additional Topic
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