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Reverse Ordering

Last updated Nov 1, 2022

# Statement 1

Let (P,)(P, \leq) be a Partial Ordering. Then the Reverse Ordering of PP, (P,)(P, \leq’) is defined as ababa \leq’ b \Leftrightarrow a \geq b a,bT\forall a,b \in T. It is a Partial Ordering.

# Proof

For all aPa \in P, aaa \leq’ a since aaa \geq a \checkmark. Suppose a,bPa,b \in P. If aba \leq’ b and bab \leq’ a, then aba \geq b and bab \geq a, so a=ba = b \checkmark. Suppose a,b,cPa,b,c \in P. If aba \leq’ b and bcb \leq’ c, then aba \geq b and bcb \geq c, so aca \geq c, implying that aca \leq’ c \checkmark. \blacksquare

# Statement 2

The Reverse Ordering of a Total Ordering is a Total Ordering.

# Proof

This follows because we can compare any a,bTa,b \in T with \leq’ using the corresponding comparison with \leq. \blacksquare