Abstract
We introduce the notions of a band category and of a weakly orthodox category over a band. Our focus is to describe a class of weakly \(B\)-orthodox semigroups, where \(B\) denotes a band of idempotents. In particular, we investigate orthodox semigroups, by using orthodox groupoids. Weakly \(B\)-orthodox semigroups are analogues of orthodox semigroups, where the relations \(\widetilde{\mathcal {R}}_B\) and \(\widetilde{\mathcal {L}}_B\) play the role that \({\mathcal {R}}\) and \(\mathcal {L}\) take in the regular case. We show that the category of weakly \(B\)-orthodox semigroups and admissible morphisms is equivalent to the category of weakly orthodox categories over bands and orthodox functors. The same class of weakly \(B\)-orthodox semigroups was studied in an earlier article by Gould and the author using generalised categories. Our approach here is more akin to that of Nambooripad. The significant difference in strategy is that it is more convenient to consider categories equipped with pre-orders, rather than with partial orders.
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Acknowledgments
The author was supported by charitable sponsors in Hong Kong, the National Natural Science Foundation of China (Grant No:10971160) and the Annie Curry Williamson scholarships, the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. She would like to take this opportunity to thank charitable sponsors in Hong Kong for supporting her to study in the University of York. She would like to thank Dr. Philip Wu and Ms. Catherine Hung in particular. She should like to thank Prof. K.P. Shum and Prof. X.M. Ren for, among many things, their greatest help and encouragement all the time. She would like to thank Prof. Victoria Gould for carefully reading and revising this paper and for providing useful suggestions.
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Wang, Y. Weakly \(B\)-orthodox semigroups. Period Math Hung 68, 13–38 (2014). https://doi.org/10.1007/s10998-014-0023-6
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DOI: https://doi.org/10.1007/s10998-014-0023-6