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Injective Hulls are Completions of Ordered Algebras

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Abstract

It is well-known that the Dedekind-MacNeille completion of a poset is its injective hull. We prove that the injective hull of an ordered universal algebra with respect to a specific class of monomorphisms has properties that are similar to the properties of the Dedekind-MacNeille completion of a poset. In particular, this injective hull induces a reflector functor from the category of ordered algebras and continuous morphisms to the category of sup-algebras.

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Acknowledgements

Research of X. Zhang and J.J. Feng was supported by the Guangdong Basic and Applied Basic Research Foundation, China, No. 2020A1515010206 and No. 2021A1515010248, the Science and Technology Program of Guangzhou, China, No. 202102080074. Research of V. Laan was supported by the Estonian Research Council grant PRG1204.

We would like to thank Ülo Reimaa, Kalle Kaarli and Mart Abel for useful comments. We are also grateful to the referee for many useful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, 510631, Guangzhou, China

    Xia Zhang & Jianjun Feng

  2. Institute of Mathematics and Statistics, University of Tartu, 51009, Tartu, Estonia

    Valdis Laan

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  1. Xia Zhang
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  2. Valdis Laan
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  3. Jianjun Feng
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Correspondence to Xia Zhang.

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Zhang, X., Laan, V. & Feng, J. Injective Hulls are Completions of Ordered Algebras. Order (2023). https://doi.org/10.1007/s11083-023-09645-7

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