^{}Department of Mathematics, Persian Gulf University, Bushehr, 75169, Iran

Abstract

By considering the notion of MV-algebras, we recall some results on enumeration of MV-algebras and we carry out a study on characterization of MV-algebras of orders $2$, $3$, $4$, $5$, $6$ and $7$. We obtain that there is one non-isomorphic MV-algebra of orders $2$, $3$, $5$ and $7$ and two non-isomorphic MV-algebras of orders $4$ and $6$.

[1] L. P. Belluce and A. Di Nola, Yosida type representation for perfect MV-algebras, Math. Logic Quarterly, Vol 42 (1996), pp. 551-563. [2] S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Text in Mathematics, Vol. 78, Springer-Verlag, New York Heidelberg Berlin, (1981). [3] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. Vol 88 (1958), pp. 467-490. [4] C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. Vol 93 No. 1 (1959), pp. 74-80. [5] R. Cignoli, I. D’Ottaviano and D. Mundici, Algebras das logicas the Lukasiewics, 1st ed., Centro de Logica, Epistemologia e Historia da Ciencia, Campinas, Brazil, (1994). [6] L. C. Ciungu, Directly indecomposable residuated lattices, Iranian journal of fuzzy systems Vol. 6 No. 2 (2009), pp. 7-18. [7] A. Di Nola, One chain generated varieties of MV-algebras, J. of algebra, Vol. 225 (2000), pp. 667-697. [8] A. Di Nola, R. Grigolia and A. Lettieri, Projective MV-algebras, Internat. J. Approx. Reason. Vol. 47 (2008), pp. 323-332. [9] A. Filipoiu, G. Georgescu and A. Lettieri, Maximal MV-algebras, Mathware and soft computing, Vl. 4 (1997), pp. 53-62. [10] G. Gratzer, Lattice theory First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., (1971). [11] J. Jakubik, Direct product decompositions of MV-algebras, Czech. Math. J. 44 (1994) 725-739. [12] J. Jakubik, Direct product decompositions of pseudo MV-algebras, Archivum Math. Vol. 37 (2001), pp. 131-142. [13] W. Komori, Super-Lukasiewicz propositional logics, Nagoya Math. J. Vol. 84 (1981), pp. 119-133. [14] D. Mundici, Interpretation of AFC*-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. Vol. 65 (1986), pp. 15-63. [15] S. Rasouli, B. Davvaz, Roughness in MV-algebras, Information Sciences, Vol. 180, No. 5 (2010), pp. 737-747. [16] S. Rasouli, B. Davvaz, Homomorphism, Ideals and Binary Relations on Hyper-MV Algebras, Multiple- valued Logic and Soft Computing, Vol. 17, No. 1 (2011), pp. 47-68. [17] B. Teheux, Lattice of subalgebras in the finitely generated varieties of MV-algebras, Discrete Mathematics, Vol. 307 (2007), pp. 2261-2275. [18] E. Turunen, Mathematics behind fuzzy logic. Physica-Verlag, Heidelberg, (1999).