ACTA issues

## Focal Baer semigroups and a restricted star order

J?nis C??rulis

Acta Sci. Math. (Szeged) 85:1-2(2019), 97-117
69/2017

 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.)