A cyclically ordered set is a set G with a ternary relation C which satisfies
(x,y,z), (x,z,u) (transitivity),
x,y,z, (x,y,z) or (z,y,x) (linearity).
A cyclically ordered group is a group such that G is a cyclically ordered set and it holds for any a,b. A cut on a cyclically ordered set G is a linear order on G such that . Such a cut is called regular if either contains a least element or has neither a least nor a greatest element. The set C(G) of all regular cuts on G with naturally defined cyclic order is called a completion of G.
Let , be cyclically ordered groups such that with the induced cyclic order and is a subgroup of . Then is called an extension of . The set of all extensions of is (partially) ordered by set inclusion; its greatest element is called a completion of . The authors give a (challenging) construction of a completion of a cyclically ordered group. Also, they derive necessary and sufficient conditions under which a given cut belongs to a completion of a cyclically ordered group , and a necessary and sufficient condition for a cyclically group to be complete i.e. equal to its completion.