Abstract
In this paper, we discuss the related properties of some particular derivations in semihoops and give some characterizations of them. Then, we prove that every Heyting algebra is isomorphic to the algebra of all multiplicative derivations and show that every Boolean algebra is isomorphic to the algebra of all implicative derivations. Finally, we show that the sets of multiplicative and implicative derivations on bounded regular idempotent semihoops are in one-to-one correspondence.
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References
E Posner. Derivations in prime rings, Proc Amer Math Soc, 1957, 8: 1093–1100.
G Georgescu, L Leuştean, V Preotesa. Pseudo hoops, J Mult Valued Log Soft Comput, 2005, 11: 153–184.
G Grätzer. Lattice theory, W H Freeman and Company, San Francisco, 1979.
H Masoud, M Mahboobeh. Folding theory applied to Rℓ-monoids, Annals of the University of Craiova, Mathematica and Computer Science Series, 2010, 37: 9–17.
J R Büchi, T M Owens. Complemented monoids and hoops, 1975, Unpublished manuscript.
J T Wang, P F He, Y H She. Some results on derivations of MV-algebras, Appl Math J Chinese Univ, 2023, 38: 126–143.
J T Wang, T Qian, Y H She. Characterizations of obstinate filters in semihoops, Ital J Pure Appl Math, 2019, 42: 851–862.
J T Wang, Y H She, T Qian. Study of MV-algebras via derivations, An Şt Univ Ovidius Constanta, 2019, 27(3): 259–278.
M A Kologani, S Z Song, R A Borzooei, Y B Jun. Constructing some logical algebras with hoops, Mathematics, 2019, 7(12), https://doi.org/10.3390/math7121243.
M Ward, P R Dilworth. Residuated lattice, Tran Amer Math Soc, 1939, 45: 335–354.
N O Alshehri. Derivations of MV-algebras, Int J Math Math Sci, 2010, 2010, https://doi.org/10.1155/2010/312027.
P Aglianò, I M A Ferreirim, F Montagna. Basic hoops: an algebriaic study of continuous t-norms, Stud Logica, 2007, 87: 73–98.
P Hájek. Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
P F He, B Zhao, X L **n. States and internal states on semihoops, Soft Comput, 2017, 21: 2941–2957.
P F He, X L **n, J M Zhan. On derivations and their fixed point sets in residuated lattices, Fuzzy Sets Syst, 2016, 303: 97–113.
R A Borzooei, O Zahiri. Some results on derivations of BCI-algebras, Sci Math Jpn, 2013, 26: 529–545.
R A Borzooei, M Aaly Kologani. Local and perfect semihoops, J Intell Fuzzy Syst, 2015, 29: 223–234.
S Ghorbain, L Torkzadeh, S Motamed. (⊙, ⊕)-derivations and (⊖, ⊙)-derivations on MV-algebras, Iran J Math Sci Inform, 2013, 8: 75–90.
U Höhle. Commutative residuated ℓ-monoids, In: Non-classical logics and their application to fuzzy subsets, Kluwer Academic Publishers, 1995, 32: 53–106.
X L **n. The fixed set of a derivation in lattices, Fixed Point Theory Appl, 2012, 2012: 218.
X L **n, T Y Li, J H Lu. On derivations of lattices, Inf Sci, 2008, 178: 307–316.
Y B Jun, X L **n. On derivations of BCI-algebras, Inf Sci, 2004, 59: 167–176.
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Supported by the National Natural Science Foundation of China(12271319).
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Wang, M., Zhang, Xh. Characterizations of semihoops based on derivations. Appl. Math. J. Chin. Univ. 39, 291–310 (2024). https://doi.org/10.1007/s11766-024-4386-z
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DOI: https://doi.org/10.1007/s11766-024-4386-z