Describes an involution, which is an operator that satisfies f(f(x)) = x (rule i).

If a multiplicative semigroup is available, it describes an antiautomorphism (rule ii); on a multiplicative monoid, it preserves the identity (rule iii).

If a ring is available, it should be compatible with addition (rule iv), and then defines a *-ring (see https://en.wikipedia.org/wiki/%2A-algebra ).

1. adjoint(adjoint(x)) = x ii. adjoint(x*y) = adjoint(y)*adjoint(x) (with an underlying multiplicative semigroup) iii. adjoint(1) = 1 (with an underlying multiplicative monoid) iv. adjoint(x+y) = adjoint(x)+adjoint(y) (with an underlying ring)

A *-algebra is an associative algebra A over a commutative *-ring R, where A has an involution as well. It satisfies, for x: A, y: A and r: R

v. adjoint(r * x + y) = adjoint(r)*adjoint(x) + adjoint(y)