ACTA issues

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

Jānis Cı̄rulis

Acta Sci. Math. (Szeged) 86:3-4(2020), 359-403

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.

DOI: 10.14232/actasm-019-426-3

AMS Subject Classification (1991): 20M10; 06A06, 06C15, 20M25, 16U99, 16W99

Keyword(s): Baer semigroup, D-semigroup, D-ring, generalized orthomodular poset, orthomodular lattice, relatively orthocomplemented poset, Rickart ring, right normal band, right star order, $U$-semiabundant semigroup

received 26.9.2019, revised 22.5.2020, accepted 25.5.2020. (Registered under 926/2019.)