Symmetric bi-derivations of residuated lattices. (English) Zbl 07854529
Summary: In this paper, we introduced the notion of symmetric bi-derivations on residuated lattices and investigated some related properties. Some relationships between symmetric bi-derivation and \(k\)-isotone, \(k\)-contractive and \(k\)-ideal symmetric bi-derivations are given. Also, we introduce the sets of \(k\)-fixed points of a symmetric bi-derivation and its structure is studied. In particular, we show that the “family” of sets of \(k\)-fixed points forms a residuated lattice.
MSC:
| 06C05 | Modular lattices, Desarguesian lattices |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
References:
| [1] | Çeven, Y., Symmetric bi-derivations of lattices, Quaest. Math., 32, 241-245, 2009 · Zbl 1184.06002 · doi:10.2989/QM.2009.32.2.6.799 |
| [2] | He, P.; Xin, X.; Zhan, J., On derivations and their fixed point sets in residuated lattices, Fuzzy Sets Syst., 303, 97-113, 2016 · Zbl 1403.03135 · doi:10.1016/j.fss.2016.01.006 |
| [3] | Jun, YB; Xin, XL, On derivations of BCI-algebras, Inf. Sci., 159, 167-176, 2004 · Zbl 1044.06011 · doi:10.1016/j.ins.2003.03.001 |
| [4] | Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Logic Without Contraction (2001) |
| [5] | Muhiuddin, G., On \(( , \tau )\)-derivations of BCI-algebras, Southeast Asian Bull. Math., 43, 5, 715-723, 2019 · Zbl 1449.06033 |
| [6] | Muhiuddin, G., Regularity of generalized derivations in BCI-algebras, Commun. Korean Math., 31, 2, 229-235, 2016 · Zbl 1339.06024 · doi:10.4134/CKMS.2016.31.2.229 |
| [7] | Muhiuddin, G.; Al-roqi, AM; Jun, YB; Ceven, Y., On symmetric left bi-derivations in BCI-algebras, Int. J. Math. Math. Sci., 2013 · Zbl 1279.06013 · doi:10.1155/2013/238490 |
| [8] | Maksa, GY, A remark on symmetric biadditive functions having nonnegative diagonalization, Glas. Math., 15, 35, 279-282, 1980 · Zbl 0463.39009 |
| [9] | Maks, GY, On the trace of symmetric bi-derivations, C. R. Math. Rep. Acad. Sci. Can., 9, 303-307, 1989 · Zbl 0639.39008 |
| [10] | Ozturk, MA; Sapancy, M., On generalized symmetric bi-derivations in prime rings, East Asian Math. J., 15, 2, 165-176, 1999 |
| [11] | Sapancy, M.; Ozturk, MA; Jun, YB, Symmetric bi-derivations on prime rings, East Asian Math. J., 15, 1, 105-109, 1999 |
| [12] | Posner, E., Derivations in prime rings, Proc. Am. Math. Soc., 8, 1093-1100, 1957 · Zbl 0082.03003 · doi:10.1090/S0002-9939-1957-0095863-0 |
| [13] | Turumen, E., Mathematics Behind Fuzzy Logic, 1999, Wurzburg: Physica-Verlag, Wurzburg · Zbl 0940.03029 |
| [14] | Vukman, J., Symmetric bi-derivations on prime and semi-prime rings, Aequ. Math., 38, 245-254, 1989 · Zbl 0687.16031 · doi:10.1007/BF01840009 |
| [15] | Vukman, J., Two results concerning symmetric bi-derivations on prime rings, Aequ. Math., 40, 181-189, 1990 · Zbl 0716.16013 · doi:10.1007/BF02112294 |
| [16] | Ward, P.; Dilworth, RM, Residuated lattice, Trans. Am. Math. Soc., 45, 335-354, 1939 · JFM 65.0084.01 · doi:10.1090/S0002-9947-1939-1501995-3 |
| [17] | Xin, XL; Li, TY; Lu, JH, On derivations of lattices, Inf. Sci., 1778, 307-316, 2008 · Zbl 1130.06002 · doi:10.1016/j.ins.2007.08.018 |
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