Semigroup Forum

, Volume 97, Issue 2, pp 307–324 | Cite as

Restriction \(\omega \)-semigroups

  • Yanhui WangEmail author
  • Dilshad Abdulkadir


The purpose of this paper is to investigate restriction \(\omega \)-semigroups. Here a restriction \(\omega \)-semigroup is a generalisation of an inverse \(\omega \)-semigroup. We give a description of a class of restriction \(\omega \)-semigroups, namely, restriction \(\omega \)-semigroups with an inverse skeleton. We show that a restriction \(\omega \)-semigroup with an inverse skeleton is an ideal extension of a \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse \(\omega \)-semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup with an inverse skeleton, conversely, every \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup with an inverse skeleton arises in this way.


Bruck–Reilly semigroups \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroups Inverse skeleton 



We would like to thank the referee for some suggestions about the definition of restriction semigroups, the literary background and Example 5.1. The authors would also like to thank Victoria Gould for her suggestions and comments on their manuscript.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.LeedsUK

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