Relatively Orthocomplemented Skew Nearlattices in Rickart Rings

Abstract

A class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset (R, ≤) is relatively orthocomplemented; in particular, every initial segment in it is orthomodular.

Keywords

• orthogonality
• relatively orthocomplemented poset
• restrictive semigroup
• Rickart ring
• right normal band
• right-star order
• skew nearlattice