Abstract.
Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We show that GMV-algebras can be considered as special cases of Grishin algebras. As MV-algebras are algebraic models of the Łukasiewicz logic and Grishin algebras have the analogous role for the classical bilinear logic, GMV-algebras correspond to a non-commutative logic between the above logics. Further, by A. Dvurečenskij, any GMV-algebra is isomorphic to an interval of an l-group, which in general is not commutative. This generalizes D. Mundici's representation of MV-algebras by means of intervals of abelian l-groups. In the paper (using this representation) we describe the properties of prime ideal spectra of GMV-algebras and of their factor algebras and ideals and prove that the spectrum of closed ideals of any GMV-algebra is homeomorphic to that of a completely distributive GMV-algebra.
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Received January 4, 2001; accepted in final form May 2, 2002.
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Rachunek, J. Prime spectra of non-commutative generalizations of MV-algebras. Algebra univers. 48, 151–169 (2002). https://doi.org/10.1007/PL00012447
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DOI: https://doi.org/10.1007/PL00012447