Orthosymmetric ortholattices and Rickart \(^*\)-rings (Q1907551)

An orthosymmetric ortholattice (OSOL) [\textit{R. Mayet}, Proc. Am. Math. Soc. 114, No. 2, 295-306 (1992; Zbl 0741.06006)]\ is an ortholattice equipped with a binary operation \(S\) satisfying the following axioms (where \(S(a, b)\) is denoted by \(S_a (b)\)): 1.) Every \(S_a\) is an isotony automorphism of \(L\) and \(S_a \circ S_b \circ S_a= S_{S_a (b)}\). 2.) If \(a\), \(b\) are orthogonal, then \(S_a \circ S_b= S_b \circ S_a= S_{a b}\). 3.) \(S_a (b)= b\) if and only if \(a\) and \(b\) commute. In the paper some new properties of OSOLs are proved and the lattice of projections of a Rickart *-ring satisfying the implication \((2x= 0\Rightarrow x=0)\) is presented as an OSOL.