Every synaptic algebra has the monotone square root property
  A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW*-algebra. In this paper we prove that a synaptic algebra A has the monotone square property, i.e., if a and b are positive elements, then if a is less or equal than b, then the square root of a is less or equal than the square root of b. 
David J. Foulis, Anna Jencova, Sylvia Pulmannova
math.OA