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Multiplicative and implicative derivations on residuated multilattices. (English) Zbl 1436.06014

Summary: In this paper, we extend the study of derivations on residuated lattices to residuated multilattices. Special types of derivations (implicative and multiplicative) and their connections with the complemented elements are investigated. In particular, one obtains that the good ideal derivations of a bounded residuated multilattice are completely determined by its complemented elements. Supporting examples of all the notions treated are also included.

MSC:

06B75 Generalizations of lattices
06B05 Structure theory of lattices
03G25 Other algebras related to logic
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References:

[1] Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, Columbia · Zbl 0321.06012
[2] Bell HE, Kappe LC (1989) Rings in which derivations satisfy certain algebraic conditions. Acta Math Hung 53:339-346 · Zbl 0705.16021 · doi:10.1007/BF01953371
[3] Cabrera IP, Cordero P, Gutierrez G, Martinez J, Ojeda-Aciego M (2014) On residuation in multilattices: filters, congruences, and homomorphisms. Fuzzy Sets Syst 234:1-21 · Zbl 1315.06008 · doi:10.1016/j.fss.2013.04.002
[4] Hansen DJ (1981) An axiomatic characterization of multilattices. Discret Math 33(1):99-101 · Zbl 0447.06005 · doi:10.1016/0012-365X(81)90263-6
[5] He P, Xinb X, Zhan J (2016) On derivations and their fixed point sets in residuated lattices. Fuzzy Sets Syst 303:97-113 · Zbl 1403.03135 · doi:10.1016/j.fss.2016.01.006
[6] Johnston IJ (1990) Some results involving multilattice ideals and distributivity. Discret Math 83(1):27-35 · Zbl 0716.06004 · doi:10.1016/0012-365X(90)90218-7
[7] Jun YB, Xin XL (2004) On derivations of BCI-algebras. Inf Sci 159:167-176 · Zbl 1044.06011 · doi:10.1016/j.ins.2003.03.001
[8] Martinez J, Gutiérrez G, de Guzmàn IP, Cordero P (2005) Generalizations of lattices via non-deterministic operators. Discret Math 295(1-3):107-141 · Zbl 1085.06005 · doi:10.1016/j.disc.2004.08.043
[9] Medina J, Ojeda-Aciego M, Ruiz-Calviño J (2007) Fuzzy logic programming via multilattices. Fuzzy Sets Syst 158:674-688 · Zbl 1111.68016 · doi:10.1016/j.fss.2006.11.006
[10] Posner E (1957) Derivations in prime rings. Proc Am Math Soc 8:1093-1100 · Zbl 0082.03003 · doi:10.1090/S0002-9939-1957-0095863-0
[11] Rachunek J, S̆alounová D (2017) Derivations on algebras of a non-commutative generalization of the Lukasiewicz logic. Fuzzy Sets Syst 333:11-16 · Zbl 1380.06010 · doi:10.1016/j.fss.2017.01.013
[12] Wang J, Jun Y, Xin X, Zou Y (2016) On derivations on bounded hyperlattices. J Math Res Appl 36(2):151-161 · Zbl 1363.06007
[13] Xin XL, Li TY, Lu JH (2008) On derivations of lattices. Fuzzy Sets Syst 178:307-316 · Zbl 1130.06002
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