ACTA issues

Focal Baer semigroups and a restricted star order

J?nis C??rulis

Acta Sci. Math. (Szeged) 85:1-2(2019), 97-117

Abstract. The goal of the paper is to transfer some order properties of star-ordered Rickart *-rings to Baer semigroups. A focal Baer semigroup $S$ is a semigroup with 0 expanded by two unary idempotent-valued operations, $\lt $ and $\rt $, such that the left (right) ideal generated by $x\lt $ (resp., $x\rt $) is the left (resp., right) annihilator of $x$. $S$ is said to be symmetric if the ranges of the two operations coincide and $p\lt = p\rt $ for every $p$ from the common range $P$. Such a semigroup is shown to be $P$-semiabundant. If it is also Lawson reduced, then $P$ is an orthomodular lattice under the standard order of idempotents, and a restricted version of Drazin star partial order can be defined on $S$. The lattice structure of $S$ under this order is shown to be similar, in several respects, to that of star-ordered Rickart *-rings.

DOI: 10.14232/actasm-017-319-5

AMS Subject Classification (1991): 20M25, 20M10, 06F99

Keyword(s): Baer semigroup, closed idempotent, orthomodular lattice, Rickart ring, Rickart *-ring, star order

Received November 6, 2017 and in final form April 14, 2018. (Registered under 69/2017.)