Abstract. In 1991, Lawson introduced three partial orders on reduced $U$-semiabundant 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 $\les _l$ 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.
AMS Subject Classification
(1991): 20M10; 06A06, 06C15, 20M25, 16U99, 16W99
generalized orthomodular poset,
relatively orthocomplemented poset,
right normal band,
right star order,
received 26.9.2019, revised 22.5.2020, accepted 25.5.2020. (Registered under 926/2019.)