F-rings
An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring in which and imply that for all . They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is negative, even though being a square. The additional hypothesis required of f-rings eliminates this possibility.
Read more about this topic: Partially Ordered Ring