Focal Baer semigroups and a restricted star order

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, ‵ and ′, such that the left (right) ideal generated by x ‵ (resp., x ′) is the left (resp....

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Published inActa scientiarum mathematicarum (Szeged) Vol. 85; no. 1-2; pp. 97 - 117
Main Author Cīrulis, J. ānis
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.01.2019
Subjects
Algebra
Functional Analysis
Mathematics
Operator Theory
orthomodular lattice
Baer semigroup
Rickart ring
20M25
star order
closed idempotent
Rickart -ring
20M10
06F99
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Summary: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, ‵ and ′, such that the left (right) ideal generated by x ‵ (resp., x ′) is the left (resp., right) annihilator of x . S is said to be symmetric if the ranges of the two operations coincide and p ‵ = p ′ 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.
ISSN:0001-6969
2064-8316
DOI:10.14232/actasm-017-319-5