On exact category of $(m, n)$-ary hypermodules

Document Type : Research Paper


1 Department of Mathematics, Payamenoor University,P.O. Box 19395-3697, Tehran, Iran.

2 Mathematics, School of Mathematics, Statistics and Computer Science, University of Tehran



We introduce and study category of $(m, n)$-ary hypermodules as a generalization of the category of $(m, n)$-modules as well as the category of classical modules. Also, we study various kinds of morphisms. Especially, we characterize monomorphisms and epimorphisms in this category. We will proceed to study the fundamental relation on $(m, n)$-hypermodules, as an important tool in the study of algebraic hyperstructures and prove that this relation is really functorial, that is, we introduce the fundamental functor from the category of $(m, n)$-hypermodules to the category $(m, n)$-modules and prove that it preserves monomorphisms. Finally, we prove that the category of $(m, n)$-hypermodules is an exact category, and, hence, it generalizes the classical case.


[1] Ameri, R., On the categories of hypergroups and hypermodules, J. Discrete Math. Sci. Cryptogr. 6 (2003), 121-132.
[2] Ameri, R. and Norouzi, M., Prime and primary hyperideales in Krasner (m; n)- hyperring, European J. Combin. 34 (2013), 379-390.
[3] Ameri, R., Norouzi, M., and Leoreanu-Fotea, V., On Prime and primary subhypermodules of (m; n)-hypermodules, European J. Combin. 44 (2015), 175-190.
[4] Anvariyeh, S.M., Mirvakili, S., and Davvaz, B., Fundamental relation on (m; n)- hypermodules over (m; n)-hyperrings, Ars combin. 94 (2010), 273-288.
[5] Belali, S., Anvariyeh, S.M., and Mirvakili, S., Free and cyclic (m; n)-hypermodules, Tamkang J. Math. 42 (2011), 105-118.
[6] Corsini, P., "Prolegemena of Hypergroup Theory", Aviani Editor, 1993.
[7] Corsini, P., and Leoreanu-Fotea, V., "Applications of Hyperstructure Theory", Kluwer Academic Publishers, 2003.
[8] Crombez, G., On (m; n)-rings, Abh. Math. Semin. Univ. Hambg. 37 (1972), 180-199.
[9] Crombez, G. and Timm, J., On (m; n)-quotient rings, Abh. Math. Sem. Univ. Hamburg 37 (1972), 200-203.
[10] Davvaz, B. and Leoreanu-Fotea, V., "Hyperring Theory and Applacations", International Academic Press, 2007.
[11] Davvaz, B. and Vougiouklis, T., n-ary hypergroups, Iran. J. Sci. Technol. Trans. A. Sci. 30 (2006), 165-174.
[12] Dörnte, W., Untersuchungen Über einen verallgemeinerten Gruppenenbegriff, Math. Z. 29 (1928), 1-19.
[13] Gladki, p. and Worytkiewicz, K., Category of hypermodules with multi-valued morphism, http:==www.math.us.edu.pl= pgladki=inedita=cathyp.pdf.
[14] Jafarzadeh, N. and Ameri, R., On the relation between categories of (m; n)-ary hypermodules and (m; n)-ary modules, Sigma J. Eng. & Nat. Sci. 9 (2018), 133-147.
[15] Leoreanu-Fotea, V., Canonical n-ary hypergroups, Ital. J. Pure Appl. Math. 24 (2008), 247-254.
[16] Leoreanu-Fotea, V. and Davvaz, B., n-hypergroups and binary relations, European J. Combin. 29 (2008), 1027-1218.
[17] Leoreanu-Fotea, V. and Davvaz, B., Roughness in n-ary hypergroups, Inform. Sci. 178 (2008), 4114-4124.
[18] Madanshekaf, M., Exact category of hypermodules, Hindawi Publishing Corporation, 31368 (2006), 1-8.
[19] Marty, F, Sur une generalization de group, in: 8iem Congres des Mathematiciens Scandinaves, Stockholm, (1934), 45-49.
[20] Mirvakili, S. and Davvaz, B., Constructions of (m; n)-hyperrings, Mat. Vesnik, 67(1) (2015), 1-16.
[21] Mirvakili, S. and Davvaz, B., Relations on Krasner (m; n)-hyperrings, European J. Combin. 31 (2010), 790-802.
[22] Mitchell, B. ,"Theory of Categories, Pure and Applied Mathematics", Academic Press, 1965.
[23] Ostadhadi-Dehkordi, S. and Davvaz, B., A note on isomorphism theorems of Krasner (m; n)-hyperrings, Arab. J. Math. 5 (2016), 103-115.
[24] Shojaei, H. Ameri, R. and Hoskova-Mayerova, S., On properties of various morphisms in the categories of general Krasner hypermodules, Ital. J. Pure Appl. Math. 39 (2018), 475-484.
[25] Shojaei, H. and Ameri, R., Some results on categories of Krasner hypermodules, J. Fundam. Appl. Sci. 8(3S) (2016), 2298–2306.
[26] Shojaei, H. and Ameri, R., Various kinds of quotient of a canonical hypergroup, Sigma J. Eng. & Nat. Sci. 9(1) (2018), 147-155.
[27] Shojaei, H. and Ameri, R., Various kinds of freeness in categories of Krasner hypermodules, to appear in Smaller Stiinifice Ale Universitatii Ovidius Constants, Seria Matematica.
[28] Vougiouklis, T.,"Hyperstructure and their Representations", Hardonic Press, 1994.