Document Type : Research Paper


1 Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran

2 Department of Mathematics, Tarbiat Modares University, Tehran, Iran


We present some connections between the max-min general fuzzy
automaton theory and the hyper structure theory. First, we introduce a hyper
BCK-algebra induced by a max-min general fuzzy automaton. Then, we study
the properties of this hyper BCK-algebra. Particularly, some theorems and
results for hyper BCK-algebra are proved. For example, it is shown that
this structure consists of different types of (positive implicative) commutative
hyper K-ideals. As a generalization, we extend the definition of this hyper
BCK-algebra to a bounded hyper K-algebra and obtain relative results.


[1] M. A. Arbib, From automata theory to brain theory, Int. J. Man-Machine Stud., 7(3) (1975),
[2] W. R. Ashby, Design for a brain, Chapman and Hall, London, 1954.
[3] R. A. Borzooei, Hyper BCK and K-algebras, Ph.D. Thesis, Department of Mathematics,
Shahid Bahonar University of Kerman, Iran, 2000.
[4] R. A. Borzooei, A. Hasankhani, M. M. Zahedi and Y. B. Jun, On hyper K-algebras, Scientiae
Mathematicae Japonica, 52(1) (2000), 113-121.
[5] A. W. Burks, Logic, biology and automata-some historical reflections, Int. J. Man-Machine
Stud., 7(3) (1975), 297-312.
[6] P. Corsini, Prolegomena of hyper group theory, Avian Editor, Italy, 1993.
[7] P. Corsini and V. Leoreanu, Applications of hyper structure theory, Advances in Mathematics,
Kluwer Academic Publishers, 5 (2003).
[8] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal
of Approximate Reasoning, 38 (2005), 175-214.
[9] J. E. Hopcroft, R. Motwani and J. D. Ullman, Introduction to automata theory, languages
and computation, seconded, Addison-Wesley, Reading, MA, 2001.
[10] M. Horry and M. M. Zahedi, Uniform and semi-uniform topology on general fuzzy automata,
Iranian Journal of Fuzzy Systems, 6(2) (2009), 19-29.
[11] M. Horry and M. M. Zahedi, Hypergroups and general fuzzy automata, Iranian Journal of
Fuzzy Systems, 6(2) (2009), 61-74.
[12] Y. Imai and K. Iseki, On axiom systems of propositional calculi, XIV Proc. Japan Academy,
42 (1966), 19-22.
[13] D. S. Malik and J. N. Mordeson, Fuzzy discrete structures, Physica-Verlag, NewY ork, 2000.
[14] F. Marty, Sur une generalization de la notion de groups, 8th Congress Math. Scand naves,
Stockholm, (1934), 45-49.
[15] W. S. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity,
Bull. Math. Biophysics., 5 (1943), 115-133.
[16] J. Meng and Y. B. Jun, BCK-algebra, Kyung Moons, Co., Seoul, 1994.
[17] M. L. Minsky, Computation: finite and infinite machines, Prentice-Hall, Englewood Cliffs,
NJ, Chapter 3, (1967), 32-66.
[18] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages, theory and applications,
Chapman and Hall/CRC, London/Boca Raton, FL, 2002.
[19] M. A. Nasr Azadani and M. M. Zahedi, S-absorbing set and (P)-decomposition in hyper
K-algebras, Italian Journal of Pure and Applied Mathematics, to appear.
[20] W. Omlin, K. K. Giles and K. K. Thornber, Equivalence in knowledge representation: automata,
runs, and dynamicfuzzy systems, Proc. IEEE, 87(9) (1999), 1623-1640.
[21] W. Omlin, K. K. Thornber and K. K. Giles, Fuzzy finite-state automata can be deterministically
encoded into recurrent neural networks, IEEE Trans. Fuzzy Systems, 5(1) (1998),
[22] T. Roodbari, Positive implicative and commutative hyper K-ideals, Ph.D. Thesis, Department
of Mathematics, Shahid Bahonar University of Kerman, Iran, 2008.
[23] T. Roodbari, L. Torkzadeh and M. M. Zahedi, Normal hyper K-algebras, Sciatica Mathematical
Japonica, 68(2) (2008), 265-278.
[24] T. Roodbari and M. M. Zahedi, Positive implicative hyper K-ideals II, Scientiae Mathematicae
Japonica, 66(3) (2007), 391-404.
[25] L. Torkzadeh, Dual positive implicative and commutative hyper K-ideals, Ph.D. Thesis, Department
of Mathematics, Shahid Bahonar University of Kerman, Iran, 2005.
[26] L. Torkzadeh, T. Roodbari and M. M. Zahedi, Hyper stabilizers and normal hyper BCKalgebras,
Set-Valued Mathematics and Applications, to appear.
[27] A. Turing, On computable numbers, with an application to the entscheidungs problem, Proc.
London Math. Soc., 42 (1936-37), 220-265.
[28] J. Von Neumann, Theory of self-reproducing automata, University of Illinois Press, Urbana,
[29] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept
to pattern classification, Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.
[30] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[31] M. M. Zahedi, R. A. Borzooei and H. Rezaei, Some classifications of hyper K-algebra of order
3, Math. Japon., (2001), 133-142.
[32] M. M. Zahedi, M. Horry and K. Abolpor, Bifuzzy (general) topology on max-min general
fuzzy automata , Advances in Fuzzy Mathematics, 3(1) (2008), 51-68.