## Thursday, July 25, 2013

### Algebraic preorders

Two of the most common algebraic structures, binary relations and monoids, have a preorder associated with them:
• Monoids: reachability can be defined by $(x <= y) \implies \exists \; a,b: axb = y$
• Binary relations: reachability can be defined by $x <= y$ if there exists some path from x to y
I believe that the applicability of order theory to such common structures is a solid justification for its foundational status.