b06c90a2-1355-4ec3-a13d-45d8e1dd325920231006031820291wseas:wseasmdt@crossref.orgMDT DepositPROOF2732-99412944-916210.37394/232020https://wseas.com/journals/proof/index.php22020232202023310.37394/232020.2023.3https://wseas.com/journals/proof/2023.phpMahsaDavodianDepartment of Mathematics, University of Golestan, Gorgan, IRANMohsenAsghari-LarimiDepartment of Mathematics, University of Golestan, Gorgan, IRANRezaAmeriDepartment of Mathematics, Faculty of Basic Science, University of Tehran, IRANThe aim of this paper is the study directly indecomposible multialgebras. In this regards, first the isomorphism theorems and correspondence theorem for multialgebras. Then by applying congruences relation on multialgebras factor multialgebras are constructed and some important properties of them are obtained. In particular, it is shown that every finite multialgebra is isomorphic to a direct products of directly indecomposable of multialgebras. Finally, subdirect products and subdirect irreducible of multialgebras are investigated and Birkoff’s theorem is extended to multialgebras.1062023106202358668https://wseas.com/journals/proof/2023/a16proof-006(2023).pdf10.37394/232020.2023.3.8https://wseas.com/journals/proof/2023/a16proof-006(2023).pdfR. Ameri, I. G. Rosenberg, Congruences of Multialgebras, J. of Multiple valued log. Soft Comput, 15(5-6) (2009) 525–536. 10.1007/3-540-45631-7_70R. Ameri, M. M. Zahedi, T-fuzzy hyperalgebraic System, In Advances in Soft ComputingAFSS 2002: 2002 AFSS International Conferences on Fuzzy Systems Calcutta, India, February 3-6, 2002 Proceedings, (2002) (pp. 509– 514). Springer Berlin Heidelberg. I. Bosnjak, R. Madarasz, On power structures, Algebra and Discrete Mathematics, (2) (2003) 14–35. P. Corsini, Prolegomena of hypergroup theory, Aviani editore, (1993). M. M. Ebrahimi, A. Karimi, M. Mahmoudi, Limits and colimits in categories in universal hyperalgebra, Algebras Groups and Geometries, (22) (2005) 169–182. M. M. Ebrahimi, A. Karimi, M. Mahmoudi, Quotients and isomorphism theorems of universal hyperalgebras, Italian journal of Pure and Applied Mathematics, (18) 9 (2005). A. K. Feizabadi, M. Mahmoudi, M. M. Ebrahimi, Representations of the fundamental relations in universal Hyperalgebras, Journal of MultipleValued Logic & Soft Computing, (21) (2013). G. Gratzer, Lattice theory: First concepts and distributive lattices, Courier Corporation, (2009). S. Mac Lane, Categories for the working mathematician, Springer Science & Business Media, (Vol. 5), (2013). F. Marty, Sur une generalization de la notion de groups, In 8th Congress Math. Scandinaves Stockholm, (1934) C. Pelea, On the direct product of multialgebras, Studia uni.Babes-bolya,mathematica, vol XLVIII, (2003)93–98. C. Pelea, On the fundamental relation of a multialgebra, Italian journal of pure andapplid mathematics, Vol. 10, (2001)141–146. C. Pelea, Identities and multialgebras, Italian journal of pure andapplid mathematics, Vol. 21, (1967)327–342. 10.4064/fm-56-2-223-240H. E. Pickett, Subdirect representations of relational systems, Fundamenta Mathematicae, 2(56) (1964), 223–240. 10.1007/978-1-4613-8130-3H. P. Sankappanavar, & S. Burris, A course in universal algebra, Graduate Texts Math, (1981), 78–56. 10.1016/0012-365x(85)90145-1D. Schweigert, Congruence relations of multialgebras, Discrete Mathematics, 53 (1985), 249– 253. 10.28924/2291-8639-16-2018-793H. Shojaei, & R. Ameri, Various kind of freeness in the categories of krasner hypermodules, International Journal of Analysis and Applications, 16(6) (2018), 793–808.