The -c--convexness of subgroup on some partially ordered relation
Imam Nugraha Albania, Rizky Rosjanuardi, Sumanang Muhtar Gozali

Departmen Pendidikan Matematika,
Universitas Pendidikan Indonesia.
albania[at]upi.edu

Abstract

We define the partial order on group -\mathbb{Z}\oplus\mathbb{Z}- denoted by -\preceq-. By using -\preceq-, we derive the ternary relation -R(a,b,c)- on -\mathbb{Z}\oplus\mathbb{Z}-. It is known that the -c--convex subgroup of cyclically ordered group depends on such ternary relation. Eventhough the definition of cyclically order demands totality, however, in this paper we define the relation -R- that is not total. We prove that for any non-zero integers -m- and -n-, the non-trivial -c--convex subgroups of -m\mathbb{Z}\oplus n\mathbb{Z}- with respect to such -R- are of the form -m\mathbb{Z}\oplus 0- or -0\oplus n\mathbb{Z}- and we derive its direct consequence.

Keywords: Ternary relation, -c--convex subgroups

Topic: Mathematics