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Pseudolattice properties of the star-orthogonal partial ordering for star-regular rings

It is shown that a star-regular ring R forms a pseudo upper semilattice under the star-orthogonal partial ordering. That is, for every a, b in R, the set {clc > a, c > b) is nonempty if and only if a V b exists in R, in which case a V b = a + (1aat)bb*[(l ata)b*]t.