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directoid; antitone involution; sectionally switching mapping; MV-algebra; NMV-algebra; WMV-algebra; skew MV-algebra; implication algebra
It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.
[1] Abbott J. C.: Semi-boolean algebra. Matem. Vestnik 4 (1967), 177–198. MR 0239957 | Zbl 0153.02704
[2] Chajda I., Eigenthaler G., Länger H.: Congruence Classes in Universal Algebra. Heldermann, Lemgo (Germany), 2003. MR 1985832
[3] Chajda I., Halaš R., Kühr J.: Distributive lattices with sectionally antitone involutions. Acta Sci. Math. (Szeged) 71 (2005), 19–33. MR 2160352
[4] Chajda I., Halaš R., Kühr J.: Implication in MV-algebras. Algebra Universalis 52 (2004), 377–382. DOI 10.1007/s00012-004-1862-4 | MR 2120523
[5] Chajda I., Kühr J.: A non-associative generalization of MV-algebras. (to appear). MR 2357826
[6] Chang C. C.: Algebraic analyses of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467–490. DOI 10.1090/S0002-9947-1958-0094302-9 | MR 0094302
[7] Cignoli R. L. O., D’Ottaviano I. M. L., Mundici D.: Algebraic Foundations of Many- Valued Reasoning. Kluwer Acad. Publ., Dordrecht, 2000. MR 1786097
[8] Georgescu G., Iorgelescu A.: Pseudo MV-algebras. Multiple Valued Log. 6 (2001), 95–135. MR 1817439
[9] Halaš R., Plojhar L.: Implication reducts of weak MV-algebras. Contributions to General Algebra 18, Verlag Heyn, 2007, to appear. MR 2399238
[10] Halaš R., Plojhar L.: Weak MV-algebras. (to appear). MR 2399238
[11] Ježek J., Quackenbush R.: Directoids: algebraic models of up-directed sets. Algebra Universalis 27 (1990), 49–69. DOI 10.1007/BF01190253 | MR 1025835
[12] Rachůnek J.: A non-commutative generalization of MV-algebras. Czech. Math. J. 52 (2002), 255–273. DOI 10.1023/A:1021766309509 | MR 1905434 | Zbl 1012.06012
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