Order structure of U-semiabundant semigroups and rings. Part I: Left Lawson’s order

Abstract

In 1991, Lawson introduced three partial orders on reduced Usemiabundant semigroups. Their definitions are formally similar to recently discovered characteristics of the diamond, left star and right star orders respectively on Rickart *-rings; lattice properties of these orders have been studied by several authors. Motivated by these similarities, we turn to the lattice structure of U-semiabundant semigroups and rings under Lawson’s orders. In this paper, we deal with his order 6l on (a version of) right U-semiabundant semigroups and rings. In particular, existence of meets is investigated, it is shown that (under some natural assumptions) every initial section of such a ring is an orthomodular lattice, and explicit descriptions of the corresponding lattice operations are given.