非结合MV-代数中的剩余
Residuation in non-associative MV-algebras
摘要：
每个MV-代数都可以转化为满足可除性和双重否定律的剩余格。在我们以前的论文中，我们引入了NMV-代数的概念，它是MV-代数的非结合修改。一个自然的问题是，一个NMV-代数是否也可以转换成一个剩余结构。与MV-代数相反，NMV-代数不是基于格，而是仅基于有向偏序集，并且二元运算不需要是结合的，因此我们不能期望得到剩余格，而只能得到一个本质上较弱的结构，称为条件剩余偏序集。考虑几个额外的自然条件，我们证明了每个NMV-代数可以转换成这样的结构。反过来，每个这样的结构都可以组织成一个NMV-代数。此外，我们研究一个代数的更强版本，其中二元运算甚至是单调的。我们表明，这样的代数可以组织成一个剩余偏序集，相反，每一个剩余偏序集可以在这个结构转换。
英文摘要：
It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In our previous papers we introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study a bit more stronger version of an algebra where the binary operation is even monotonous. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.
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