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1,
* a <= b implies a^* <= b^* (monotonicity),
* aⁿ <= a^* for every natural number n, where aⁿ is defined as n-fold multiplication of a,
* (a^*)(a^*) = a^*,
* (a^*)^* = a^*,
* 1 + a(a^*) = a^* = 1 + (a^*)a,
* ax = xb implies (a^*)x = x(b^*),
* ((ab)^*)a = a((ba)^*),
* (a+b)^* = a^*(b(a^*))^*, and
* pq = 1 = qp implies q(a^*)p = (qap)^*.

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