Abstract
Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.
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Acknowledgements
We thank the referee for his/her careful reading of a lengthy pair of articles, and for a number of helpful suggestions. The first and third authors would like to thank Macchiato Fax for providing us with many a Klub sendvič.
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Presented by M. Jackson.
This article is part of the topical collection “The 5th Novi Sad Algebraic Conference (NSAC 2017)” edited by P. Marković, M. Maróti and A. Tepavčević.
This work was initiated during visits of the third author to Chiang Mai in 2015, and to Novi Sad in 2016; he thanks these institutions for their generous hospitality. The first and second authors are supported by Grant Nos. 174019 and 174018, respectively, of the Ministry of Education, Science, and Technological Development of the Republic of Serbia.
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Dolinka, I., Đurđev, I., East, J. et al. Sandwich semigroups in locally small categories I: foundations. Algebra Univers. 79, 75 (2018). https://doi.org/10.1007/s00012-018-0537-5
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DOI: https://doi.org/10.1007/s00012-018-0537-5