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Restriction \(\omega \)-semigroups

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Yanhui Wang

  2. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Yanhui Wang

  3. Leeds, UK

    Dilshad Abdulkadir

Authors
  1. Yanhui Wang
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  2. Dilshad Abdulkadir
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Corresponding author

Correspondence to Yanhui Wang.

Additional information

Communicated by Mikhail Volkov.

The first author was supported by NSFC (Grant Nos. 11471255, 11501331), Shandong Province Natural Science Foundation (Grant No. BS2015SF002), SDUST Research Fund (No. 2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province.

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Wang, Y., Abdulkadir, D. Restriction \(\omega \)-semigroups. Semigroup Forum 97, 307–324 (2018). https://doi.org/10.1007/s00233-018-9961-2

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