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The annihilator of fuzzy subgroups. (English) Zbl 1423.20071
Summary: We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper, we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in Euclidean spaces. We find that in the natural algebraic duality of a group, a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator. We also study the behavior of annihilators with respect to unions and intersections. Some illustrative examples of annihilators of fuzzy subgroups are shown, both with finite and infinite rank.
##### MSC:
 20N25 Fuzzy groups 20K27 Subgroups of abelian groups 06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
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##### References:
 [1] Ajmal, N., Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups, Fuzzy Sets Syst., 61, 3, 329-339, (1994) · Zbl 0832.20086 [2] Anderson, M. E.; Feil, T. H., Lattice-Ordered Groups: An Introduction, vol. 4, (1988), Springer Science & Business Media [3] Aslam, M.; Thaheem, A. B., On certain ideals in BCK-algebras, Math. Jpn., 36, 895-906, (1991) · Zbl 0743.06014 [4] Cignoli, R.; Torrens, A., An algebraic analysis of product logic, Mult. Valued Log., 5, 45-65, (2000) · Zbl 0962.03059 [5] Cignoli, R. L.; d’Ottaviano, I. M.; Mundici, D., Algebraic Foundations of Many-Valued Reasoning, vol. 7, (2013), Springer Science & Business Media [6] Das, P. S., Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84, 1, 264-269, (1981) · Zbl 0476.20002 [7] Dey, A.; Pal, M., Properties of fuzzy inner product spaces, Int. J. Fuzzy Log. Syst., 4, 2, 21-37, (2014) [8] Di Nola, A.; Lettieri, A., Perfect MV-algebras are categorically equivalent to abelian ℓ-groups, Stud. Log., 53, 3, 417-432, (1994) · Zbl 0812.06010 [9] Diamond, P., Fuzzy least squares, Inf. Sci., 46, 3, 141-157, (1988) · Zbl 0663.65150 [10] Gerla, G., Fuzzy Logic: Mathematical Tools for Approximate Reasoning, Trends in Logic, vol. 11, (2001), Springer: Springer Netherlands · Zbl 0976.03026 [11] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis I, Grundlehren der Mathematischen Wissenschaften, vol. 115, (1979), Springer-Verlag: Springer-Verlag Berlin [12] Imai, Y.; Iséki, K., On axiom systems of propositional calculi. XIV, Proc. Jpn. Acad., 42, 6, 19-22, (1966) · Zbl 0156.24812 [13] Iséki, K., An algebra related with a propositional calculus, Proc. Jpn. Acad., 42, 1, 26-29, (1966) · Zbl 0207.29304 [14] Jun, Y. B.; Xin, X. L., Fuzzy prime ideals and invertible fuzzy ideals in BCK-algebras, Fuzzy Sets Syst., 117, 3, 471-476, (2001) · Zbl 0968.06014 [15] Jun, Y. B.; Xin, X. L., Involutory and invertible fuzzy BCK-algebras, Fuzzy Sets Syst., 117, 3, 463-469, (2001) · Zbl 0968.06015 [16] Mordeson, J. N.; Buthani, K. R.; Rosenfeld, A., Fuzzy Group Theory, Studies in Fuzziness and Soft Computing, vol. 182, (2005), Springer [17] Mundici, D., Interpretation of $$AF C^\ast$$-algebras in Łukasiewicz sentential calculus, J. Funct. Anal., 65, 1, 15-63, (1986) · Zbl 0597.46059 [18] Mundici, D., MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Jpn., 31, 889-894, (1986) · Zbl 0633.03066 [19] Sebastian, S.; Babusundar, S., Existence of fuzzy subgroups of every level-cardinality upto $$\operatorname{\aleph}_0$$, Fuzzy Sets Syst., 67, 3, 365-368, (1994) · Zbl 0844.20069 [20] Yang, M. S.; Lin, T. S., Fuzzy least-squares linear regression analysis for fuzzy input – output data, Fuzzy Sets Syst., 126, 3, 389-399, (2002) · Zbl 1006.62055
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