Opposite monoid

Definition

Let $(M,*,e)$ be a monoid (set $M$, binary operation $*$, identity element $e$). Then, the opposite monoid $M^{op}$, is defined as the monoid $(M,\cdot ,e)$, where:

$a\cdot b:=b*a$

The fact that this is still a monoid follows from the fact that the axioms of associativity and neutral element enjoy a left-right symmetry.