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A non-commutative generalization of MV-algebras. (English) Zbl 1012.06012
Summary: We define a generalization of MV-algebras as algebras \(A = (A, \oplus , \odot , \neg , \sim , 0, 1)\) of signature \(\langle 2,2,1,1,0,0\rangle \) in which the binary operations \(\oplus \) and \(\odot \) in general need not be commutative. (MV-algebras as introduced by Chang are then called commutative.) In the first part of the paper some properties of noncommutative MV-algebras are described. In the second part it is proved that (noncommutative) MV-algebras are in a one-to-one correspondence with some bounded noncommutative \(DRl\)-monoids, and in the last section it is shown that every interval \([0, u]\) of any (noncommutative) \(l\)-group can be viewed as an MV-algebra and that every linearly ordered MV-algebra is isomorphic to an analogous interval of some linearly ordered loop.

MSC:
06D35 MV-algebras
06F05 Ordered semigroups and monoids
06F15 Ordered groups
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