# Addition and Multiplication of Well Orderings

• March 8th 2009, 12:32 AM
Treblesteph
Addition and Multiplication of Well Orderings
Show if $\left \langle A , R \right \rangle , \left \langle B , S \right \rangle \in W O$ then the sum $\left \langle C , T \right \rangle \in W O$
Where $a T b \Leftrightarrow ( a \in A \wedge b \in B ) \vee ( a , b \in A \wedge a R b ) \vee ( a , b \in B \wedge a S b )$
Show if $\left \langle A , R \right \rangle , \left \langle B , S \right \rangle \in W O$ then the product $\left \langle C , U \right \rangle \in W O$
Where $\left \langle x , y \right \rangle U \left \langle x ^{\prime} , y ^{\prime} \right \rangle \Leftrightarrow ( y S y ^{\prime} ) \vee ( y = y ^{\prime} \wedge x R x ^{\prime} )$