Relation between hyper $K$-algebras with superlattices and hypersemilattices

Document Type : Original Article


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

2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran


In this paper, by considering the concepts of hypersemilattice and superlattice, we prove that any commutative and positive implicative hyper $K$-algebra, is a hypersemilattice. Moreover, we prove that any bounded commutative hyper $K$-algebra with some con


[1] Z. Bin, X. Ying, H.S. Wei, Hypersemilattices, Chinese Scientific Papers Online, 10 pages,
[2] R.A. Borzooei, W.A. Dudek, N. Koohestani, On hyper BCC-algebras, Hindawi Publishing
Corporation International Journal of Mathematics and Mathematical Sciences, 2006 (2006),
[3] R.A. Borzooei, A. Hasankhani, M.M. Zahedi, Y.B. Jun, On hyper K-algebras, Math Japonica,
1 (2000), 113–121.
[4] X.Z. Guo, X.L. Xin, Hyperlattices, Pure and Applied Mathematics, 20 (2004) 40–43.
[5] 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.
[6] F. Marty, Sur une generalization de la notion degroups, 8th Congress of Mathematics Scandinaves,
Stockholm, (1934), 45–49.
[7] J. Mittas, M. Konstantinidou, Sur une nouvelle g´en´eration de la notion de treillis. Les
supertreillis et certaines de leurs proprites generales, Annales Sciences Mathématiques Blaise
Pascal, Series Mathematics, 25 (1989), 61-83.
[8] T. Roodbari, L. Torkzadeh, M.M. Zahedi, Simple hyper K-algebras, Quasigroups and Related
Systems, 16 (2008), 131–140.
[9] X.L. Xin, Hyper BCI-algebras, Discussiones Mathematicae General Algebra and Applications,
26 (2006), 5–19.
[10] M.M. Zahedi, R.A. Borzooei, H. Hasankhani, Some results on hyper K-algebras, Scientiae
Mathematicae, 3(1) (2000), 53–59