Sorting the pieces

Sorting is the first act of mathematics: choose a property, then group by it. The pieces can be sorted many ways. By how they move: rook, bishop, and queen slide in straight lines, the knight jumps, the king and pawn step one square. By colour: black or white. By value: pawn $1$, knight $3$, bishop $3$, rook $5$, queen $9$.

A sorting rule is a function from pieces to categories, and a good rule puts each piece in exactly one group. Changing the rule changes the groups, but the pieces stay the same. Ordering is sorting by a rule that also ranks the groups, such as increasing value.