Janowitz proved in 1983 that the initial segments of a Rickart *-ring with the star order are orthomodular posets. In this paper, the same result is proved for the leftstar order * ≤, which was introduced by Marovt et al., by finding an orthogonality which corresponds to * ≤ in a certain way and then applying a result proved by C¯ırulis which states that the initial segments of any quasi-orthomodular set are orthomodular.
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