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.