Opposite monoid

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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.