Implications pairs, adjoint pairs and adjoint triples provide general residuated structures considered in different mathematical theories. In this paper, we carry out a deep study on the operators involved in these structures, showing how they are characterized by means of the irreducible elements of a complete lattice. Moreover, the structure of each class of these operators will be analyzed. As a consequence, the use of these operators in real problems will be more tractable, fostering their consideration as basic and useful operators for providing, for instance, preferences among attributes and objects in a given database.

中文翻译：

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伴随三元组的代数结构和表征

蕴涵对、伴随对和伴随三元组提供了不同数学理论中考虑的一般残差结构。在本文中，我们对这些结构中涉及的算子进行了深入研究，展示了它们是如何通过完整晶格的不可约元素来表征的。此外，将分析这些运算符的每一类的结构。因此，在实际问题中使用这些算子将更容易处理，促进将它们视为基本和有用的算子，例如提供给定数据库中属性和对象之间的偏好。