2018/10/04 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1
Mathematics · #03G10 #06A11 #06D35 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03G10 #msc:06A11 #msc:06D35
paper · pdf · doi:10.48550/arxiv.1810.02405
arxiv created 2018/10/04 · arxiv updated 2018/10/08
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.