×

zbMATH — the first resource for mathematics

On interval subalgebras of generalized MV-algebras. (English) Zbl 1141.06006
Every interval \([a,b]\) of a GMV-algebra \(A\) is a GMV-algebra whose operations are defined as polynomials of \(A\) (see [I. Chajda and J. Kühr, “GMV-algebras and meet-semilattices with sectionally antitone permutations”, Math. Slovaca 56, 275–288 (2006; Zbl 1141.06002)]). The present paper gives an alternative proof using the representation of \(A\) via the functor \(\Gamma \). Specifically, if \(A=\Gamma (G,u)\) for a unital \(\ell \)-group \((G,u)\), then every interval \([a,b]\) of \(A\) is equipped with operations inherited from \(G\) making \([a,b]\) into a GMV-algebra. It is also shown that \([a,b]\) is always isomorphic to an interval subalgebra of the form \([0,c]\).

MSC:
06D35 MV-algebras
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] CHAJDA I.-HALAŠ R.-KÜHR J.: Implication in MV-algebras. Algebra Universalis 52 (2004), 377-382. · Zbl 1097.06011
[2] CHAJDA I.-HALAŠ R.-KÜHR J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged) 71 (2005), 19-33. · Zbl 1099.06006
[3] CHAJDA I.-KÜHR J.: A note on interval MV-algebras. Math. Slovaca 56 (2006), 47-52. · Zbl 1164.06010
[4] CHAJDA I.-KÜHR J.: GMV-algebras and meet-semilattices with sectionally antitone permutations. Math. Slovaca 56 (2006), 275 288. · Zbl 1141.06002
[5] CIGNOLI R.-D’OTTAVIANO M. L.-MUNDICI D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0937.06009
[6] DVUREČENSKIJ A.: Pseudo \(MV\)-algebras are intervals in \(\ell\)-groups. J. Aust. Math. Soc. 72 (2002), 427-445. · Zbl 1027.06014
[7] GEORGESCU G.-IORGULESCU A.: Pseudo MV-algebras: a noncommutative extension of MV-algebras. The Proceedings of the Fourth International Symposium on Economic Informatics, INFOREC Printing House, Bucharest, 1999, pp. 961-968. · Zbl 0985.06007
[8] GEORGESCU G.-IORGULESCU A.: Pseudo MV-algebras. Mult.-Valued Log. (Special issue dedicated to Gr. C Moisil) 6 (2001), 95-135. · Zbl 1014.06008
[9] JAKUBÍK J.: Direct product decompositions of pseudo MV-algebras. Arch. Math. (Brno) 37 (2001), 131-142. · Zbl 1070.06003
[10] JAKUBÍK J.: On intervals and the dual of a pseudo MV-algebra. Math. Slovaca 56 (2006), 213-221. · Zbl 1150.06013
[11] RACHŮNEK J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52 (2002), 255-273. · Zbl 1012.06012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.