Residuation algebras with functional duals

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.

Original language	English
Article number	40
Pages (from-to)	1-10
Number of pages	10
Journal	Algebra Universalis
Volume	80
Issue number	4
Early online date	10 Sept 2019
DOIs	https://doi.org/10.1007/s00012-019-0613-5
Publication status	Published - Dec 2019

Funding

Funders	Funder number
Horizon 2020 Framework Programme	670624

Keywords

Canonical extensions
Definability of functionality
Residuation algebras

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10.1007/s00012-019-0613-5

Fussner-Palmigiano2019_Article_ResiduationAlgebrasWithFunctionalDualsFinal published version, 239 KBLicence: Article 25fa Dutch Copyright Act

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AB - 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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KW - Definability of functionality

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