Constructive dualities have been recently proposed for some lattice based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining "choice-free spatial dualities for other classes of algebras [$\ldots$], giving rise to choice-free completeness proofs for non-classical logics''. We present in this article a way to complete the Holliday-Bezhanishvili project (uniformly, for any normal lattice expansion) by recasting recent relational representation and duality results in a choice-free manner. These results have some affinity with the Moshier and Jipsen duality for bounded lattices with quasi-operators, except for aiming at representing operators by relations, extending the J\'{o}nsson-Tarski approach for BAOs, and Dunn's follow up approach for distributive gaggles, to contexts where distribution may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases.

翻译：近年来，构造性对偶性被提出用于某些基于格的代数，霍利迪（Holliday）与别扎尼什维利（Bezhanishvili）提出了相关研究计划，旨在为“其他代数类建立无选择空间对偶性，从而为非经典逻辑提供无选择完备性证明”。本文通过以无选择方式重塑近期关系表示与对偶性结果，提出一种（对任意正规格扩张统一适用的）方法以完成霍利迪-别扎尼什维利计划。这些结果与Moshier和Jipsen关于带拟算子有界格的对偶性存在一定关联，其主要区别在于：本研究旨在通过关系表示算子，将Jónsson-Tarski方法（用于布尔代数带算子）及Dunn的后续方法（用于分配性团）扩展到无需假定分配律的语境中。为阐明这一框架，我们将其应用于带有某种（拟）补算子形式的格（及其逻辑），为带有最小拟补、伽罗瓦拟补、对合格（包括德摩根代数作为特例）、正交格及布尔代数的系统建立了关系框架下的典范扩展与无选择对偶性。