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Completion of a cyclically ordered group. (English) Zbl 0624.06021

A cyclically ordered set is a set G with a ternary relation C which satisfies

(x,y,z)C(z,y,x)C (asymmetry),

(x,y,z)C(y,z,x)C (cyclicity),

(x,y,z)C, (x,z,u)C(x,y,u)C (transitivity),

x,y,zG, xyzxeither (x,y,z)C or (z,y,x)C (linearity).

A cyclically ordered group is a group (G,+) such that G is a cyclically ordered set and it holds (x,y,z)C(a+x+b,a+y+b,a+z+b)C for any a,bG. A cut on a cyclically ordered set G is a linear order < on G such that x<y<z(x,y,z)C. Such a cut is called regular if (G,<) 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 (G,+), (G 1 ,+ 1 ) be cyclically ordered groups such that G 1 C(G) with the induced cyclic order and (G,+) is a subgroup of (G 1 ,+ 1 ). Then (G 1 ,+ 1 ) is called an extension of (G,+). The set of all extensions of (G,+) is (partially) ordered by set inclusion; its greatest element is called a completion of (G,+). 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 (G,+), and a necessary and sufficient condition for a cyclically group (G,+) to be complete i.e. equal to its completion.

Reviewer: V.Novák

MSC:
06F15Ordered groups
20F60Ordered groups (group aspects)
06B23Complete lattices, completions