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Definition 1. We say that the subsemigroup $S$ of the group $G$ (with the operation denoted multiplicatively) defines an order for the group $G$ , if
- $a^{-1}Sa \subseteq S \quad \forall a\in G,$
- $G = S\cup \{1\} \cup S^{-1}$ where $S^{-1} = \{s^{-1}: \,s\in S\}$ and the members of the union are pairwise disjoint.
The order ``$<$ '' of the group $G$ is explicitly given by setting in $G$ : $$a < b \,\, \Leftrightarrow \,\,ab^{-1}\in S$$ Then we speak of the ordered group $(G,\,<)$ , or simply $G$ .
Theorem 1 The order `` $<$ '' defined by the subsemigroup $S$ of the group $G$ has the following properties.
- For all $a,\,b\in G$ , exactly one of the conditions $a < b,\,\,a = b,\,\,b < a$ holds.
- $a < b \,\land\, b < c \,\,\Rightarrow\,\,a < c$
- $a < b \,\,\Rightarrow\,\, ac < bc \,\land\, ca < cb$
- $a < b \,\land\, c < d \,\,\Rightarrow\,\, ac < bd$
- $a < b \,\,\Leftrightarrow\,\, b^{-1} < a^{-1}$
- $a < 1 \,\,\Leftrightarrow\,\, a\in S$
Definition 2. The set $G$ is an ordered group equipped with zero 0, if the set $G^*$ of its elements distinct from its element 0 forms an ordered group $(G^*,\,<)$ and if
- $0a = a0 = 0 \quad\forall a\in G,$
- $0 < a \quad\forall a\in G^*.$
Cf. 7 in examples of semigroups.
- 1
- EMIL ARTIN: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
- 2
- PAUL JAFFARD: Les systèmes d'idéaux. Dunod, Paris (1960).
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