The so-called non-associative MV-algebras were introduced recently by the first author and J. Kühr in order to have an appropriate tool for certain logics used in expert systems where associativity of the binary operation is excluded, see, e.g., Botur and Halaš (Arch Math Log 48:243-255, 2009). Since implication is an important logical connective in practically every propositional logic, in the present paper we investigate the implication reducts of non-associative MV-algebras. We also determine their structures based on the underlying posets. The natural question when a poset with the greatest element equipped with sectional switching involutions can be organized into an implication NMV-algebra is solved. Moreover, congruence properties of the variety of implication NMV-algebras with, respectively, without zero are investigated. Analogously to classical propositional logic, we introduce a certain kind of Sheffer operation and we obtain a one-to-one correspondence between NMV-algebras and certain algebras built up by a Sheffer-like operation together with a unary operation.

中文翻译：

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从非关联MV代数派生的运算和结构。

第一作者和J.Kühr最近引入了所谓的非关联MV代数，以便为排除了二进制运算的关联性的专家系统中的某些逻辑使用适当的工具，例如，参见Botur和哈拉斯（Arch Math Log 48：243-255，2009）。由于蕴涵在几乎所有命题逻辑中都是重要的逻辑连接词，因此在本文中，我们研究了非关联MV代数的蕴涵约简。我们还根据潜在的姿势确定它们的结构。解决了具有最大元素且具有分段切换对合的姿态集可以组织为蕴涵NMV代数的自然问题。此外，研究了分别具有零且不为零的各种蕴涵NMV代数的同余性质。