Fragments of quasi-Nelson: residuation
Fragments of quasi-Nelson: residuation
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准纳尔逊的碎片：残留
DOI：
10.1080/11663081.2023.2203312
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发表时间：
2023
影响因子：
--
通讯作者：
U. Rivieccio
中科院分区：
文献类型：
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作者：
U. Rivieccio
拟Nelson逻辑(QNL)是最近提出的具有强否定的直觉主义逻辑和Nelson构造性逻辑的共同推广。作为一种子结构逻辑，QNL是完全Lambek演算在Nelson公理下的交换和弱化的公理推广，而它的代数对应是一种称为拟Nelson代数的剩余格。反过来，纳尔逊逻辑可以通过双重否定(或对合)公理得到QNL的公理推广，而直觉逻辑可以通过收缩公理得到QNL的推广。最近由作者和合作者发表的一系列论文开启了对QNL的片断的研究，这些片断对应于拟Nelson代数的子集。在本文中，我们关注的是包含形成剩余对的连接词的片段(么半群合取和所谓的强Nelson蕴涵)，从子结构逻辑的角度来看，这些片段是最有趣的。我们给出了相应代数类的拟等式(只要可能，就是等式)公理化，得到了它们的扭转表示，研究了它们的同余性质，并研究了几个值得注意的子簇。我们的结果专门针对对合的情况，给出了纳尔逊逻辑的相应片段及其代数对应的特征。
Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts.