A lattice-ordered group is a group with a partial order making it a lattice-ordered set,
such that left and right multiplication preserve the order. Writing the operation
additively, this means that 
implies both 
and 
(Fuchs 1963). The group need not be Abelian.
For example, 
with coordinatewise addition and coordinatewise order is a lattice-ordered Abelian
group. Its meet and join are
coordinatewise minimum and maximum. The additive group
of a lattice-ordered ring has this structure.
