Ordered Group Equivalences

Theorem

Let $\left({S, \circ, \preceq}\right)$ be an ordered group whose identity is $e$.

Let $x, y, z, \in S$.

Then the following are all equivalent:

$(1): \quad x \prec y$

$(2): \quad x \circ z \prec y \circ z$

$(3): \quad z \circ x \prec z \circ y$

$(4): \quad y^{-1} \prec x^{-1}$

$(5): \quad e \prec y \circ x^{-1}$

$(6): \quad e \prec x^{-1} \circ y$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 15$: Theorem $15.3$