The aim of this paper is to develop an algebraic and logical study of certain paraconsistent systems, from the family of the logics of formal inconsistency (LFIs), which are definable from the degree-preserving companions of logics of distributive involutive residuated lattices ($\textrm {dIRL}$s) with a consistency operator, the latter including as particular cases, Nelson logic ($\textsf {NL}$), involutive monoidal t-norm based logic ($\textsf {IMTL}$) or nilpotent minimum ($\textsf {NM}$) logic. To this end, we first algebraically study enriched dIRLs with suitable consistency operators. In fact, we consider three classes of consistency operators, leading respectively to three subquasivarieties of such expanded residuated lattices. We characterize the simple and subdirectly irreducible members of these quasivarieties, and we extend Sendlewski’s representation results for the case of Nelson lattices with consistency operators. Finally, we define and axiomatize the logics of three quasivarieties of $ \textrm {dIRL}$s and their corresponding degree-preserving companions that belong to the family of LFIs.

中文翻译：

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基于分布对合剩余格的形式不一致性逻辑

本文的目的是从形式不一致性逻辑 (LFI) 家族中对某些副一致系统进行代数和逻辑研究，这些逻辑可以从分布对合剩余格 ($\ textrm {dIRL}$s) 具有一致性算子，后者包括作为特殊情况的 Nelson 逻辑 ($\textsf {NL}$)、基于对合幺半 t 范数的逻辑 ($\textsf {IMTL}$) 或幂零最小值($\textsf {NM}$) 逻辑。为此，我们首先用合适的一致性算子对丰富的 dIRL 进行代数研究。事实上，我们考虑了三类一致性算子，分别导致了这种扩展剩余格的三个子准变量。我们刻画了这些准变量的简单和次直接不可约成员，我们将 Sendlewski 的表示结果扩展到具有一致性算子的 Nelson 格的情况。最后，我们定义并公理化了 $ \textrm {dIRL}$s 的三个拟变量及其对应的属于 LFI 家族的保度同伴的逻辑。