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Residuation in non-associative MV-algebras

In: Mathematica Slovaca, vol. 68, no. 6
Ivan Chajda - Helmut M. Länger


Year, pages: 2018, 1313 - 1320
MV-algebra, non-associative MV-algebra, conditional adjointness, residuated poset, directoid
About article:
It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr 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 an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.
How to cite:
ISO 690:
Chajda, I., Länger, H. 2018. Residuation in non-associative MV-algebras. In Mathematica Slovaca, vol. 68, no.6, pp. 1313-1320. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0181

Chajda, I., Länger, H. (2018). Residuation in non-associative MV-algebras. Mathematica Slovaca, 68(6), 1313-1320. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0181
About edition:
Published: 3. 12. 2018