State hyper BE-algebras

Document Type : Original Article


1 School of Science, Xi'an Aeronautical University, Xi'an, 710077, China

2 Department mathematics, northwest university, xian, china


In this paper, state operators on hyper BE-algebras (correspondingly, state hyper BE-algebras) are introduced and studied. State hyper filters are introduced and generated state hyper filters are represented in state hyper BE-algebras. Also, maximal (prime) state hyper filters are characterised and the relations between state maximal hyper filters and state prime hyper filters are discussed. Moreover, some related results of state (compatible) hyper congruence are obtained. Especially, it follows that there is an isotone bijection between all state strong regular $\circ$-reflexive hyper filter $\mathcal{F}_s(H,\xi)$ and all state compatible hyper congruence $\mathcal{C}_s(H,\xi)$ on state commutative transitive RD-hyper BE-algebra $(H,\xi)$.


[1] R.A. Borzooei, M. Aaly Kologani, An overview of hyper logical algebras, Journal of Algebraic Hyperstructures
and Logical Algebras, 1(3) (2020), 31–50
[2] R.A. Borzooei, A. Dvure˘censkij, O. Zahiri, State BCK-algebras and state-morphism BCK-algebras,
Fuzzy Sets and Systems, 244 (2014), 86–105.
[3] R.A. Borzooei, B. Ganji Saffar, R. Ameri, On hyper EQ-algebras, Italian Journal of Pure and Applied
Mathematics, 31 (2013), 77–96.
[4] X.Y. Cheng, X.L. Xin, Filter theory on hyper BE-algebras, Italian Journal of Pure and Applied Mathematics,
35 (2015), 509–526.
[5] X.Y. Cheng, X.L. Xin, P.F. He, Generalized state maps and states on pseudo equality algebras, Open
Mathematics, 16(1) (2018), 133–148.
[6] X.Y. Cheng, X.L. Xin, Y.B. Jun, Hyper equality algebras, Quantitative Logic and Soft Computing 2016,
Springer, Cham, (2017), 415–428.
[7] X.Y. Cheng, X.L. Xin, Y.W. Yang, Deductive systems in hyper EQ-algebras, Journal of Mathematical
Research with Applications, 37(2) (2017), 183–193.
[8] L.C. Ciungu, A. Dvure˘censkij, M. Hy˘cko, State BL-algebras, Soft Computing, 15(4) (2010), 619–634.
[9] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Dordrecht: Kluwer, 2003.
[10] B. Davvaz, A brief survey on algebraic hyperstructures: Theory and applications, Journal of Algebraic
Hyperstructures and Logical Algebras, 1(3) (2020), 15–29.
[11] B. Davvaz, A.D. Nezhad, M.M. Heidari, Inheritance examples of algebraic hyperstructures, Information
Sciences, 244 (2013), 180–187.
[12] T. Flaminio, F. Montagna, MV-algebras with internal states and probabilistic fuzzy logics, International
Journal of Approximate Reasoning, 50 (2009), 138–152.
[13] S. Ghorbani, A. Hasankhani, E. Eslami, Hyper MV-algebras, Set-Valued Mathematiics Applications, 1
(2008), 205–222.
[14] P.F. He, X.L. Xin, Y.W. Yang, On state residuated lattices, Soft Computing, 19(8) (2015), 2083–2094.
[15] Y.B. Jun, M.M. Zahedi, X.L. Xin, R.A. Borzooei, On hyper BCK-algebras, Italian Journal of Pure and
Applied Mathematics, 8 (2000), 127–136.
[16] F. Marty, Surune generalization de la notion de group, The 8th Congress Math., Scandinaves, Stockholm,
[17] D. Mundici, Averaging the truth-value in  Lukasiewicz logic, Studia Logica, 55(1) (1995), 113–127.
[18] A. Radfar, A. Rezaei, A. Borumand Saeid, Hyper BE-algebras, Novi Sad Journal of Mathematics, 44(2)
(2014), 137–147.
[19] A. Rezaei, A. Radfar, A. Boroumand Saeid, On commutative hyper BE-algebras, Facta Universitatis,
Series: Mathematics and Informatics, 30(4) (2015), 389–399.
[20] X.L. Xin, B. Davvaz, State operators and state-morphism operators on hyper BCK-algebras, Journal of
Intelligent and Fuzzy Systems, 29(5) (2015), 1869–1880.
[21] X.L. Xin, P.F. He, Y.L. Fu, States on hyper MV-algebras, Journal of Intelligent and Fuzzy Systems,
31(3) (2016), 1299–1309.
[22] X.L. Xin, P. Wang, States and measures on hyper BCK-algebras, Journal of Applied Mathematics, 2014
(2014), 1–8.