Abstract
We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras.
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Acknowledgements
We wish to thank Peter Jipsen for his careful reading and very useful comments on an earlier draft of this paper.
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Dedicated to Ralph Freese, Bill Lampe, and J.B. Nation.
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This article is part of the topical collection “Algebras and Lattices in Hawaii” edited by W. DeMeo.
The research of the first author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement no. 670624). The research of the second author was supported by the Vidi Grant 016.138.314 of the Netherlands Organization for Scientific Research (NWO), by the NWO Aspasia Grant 015.008.054, and by a Delft Technology Fellowship awarded in 2013.
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Fussner, W., Palmigiano, A. Residuation algebras with functional duals. Algebra Univers. 80, 40 (2019). https://doi.org/10.1007/s00012-019-0613-5
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DOI: https://doi.org/10.1007/s00012-019-0613-5