We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations - nowadays more commonly called multiple-conclusion logics - to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a congruence); the problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras.

翻译：我们扩展了Janusz Czelakowski关于蕴涵关系（现常称为多结论逻辑）的矩阵模型类闭包性质的经典工作，将其推广到非确定矩阵（Nmatrices）的框架中。通过将标准类算子推广至非确定情形，刻画了任意逻辑的Nmatrix模型。我们着重强调了这一更一般框架中出现的主要差异，特别是：利用任意相容等价关系（不必是合同关系）获取Nmatrix商的可能性；确定严格同态何时保持给定Nmatrix的逻辑的问题；像与原像操作不可交换这一事实，决定了从任意完备语义出发，生成一个逻辑的所有Nmatrix模型类的算子精确序列。另一方面，许多结果可顺利推广至非确定框架：例如，我们证明一个逻辑是有限基的，当且仅当其Nmatrix模型类及其补集均在超积下封闭。最后，我们提及了将抽象代数逻辑方法推广至Nmatrix诱导的逻辑以及相关非确定代数上的等式推理的可能发展方向。