(M)

by:

([*M*, ∗, *e*]) := *M* (*M* is a set, ∗ : *M* × *M* → *M* is an operation on *M*, *e* ∈ *M* such that (*M*, ∗, *e*) forms a monoid)

We write “let *a* ∈ M” for “let *a* ∈ (M).”

Remarks.

This definition retrieves a representative carrier set from a monoid (which is actually an equivalence class of monoids), and enables typical abuse of notation.