# Topics in topological MI-groups

Document Type : Research Paper

Authors

1 Department of Mathematics, Yazd University, Yazd, Iran.

2 Department of mathematics, Yazd University, Yazd, Iran

10.29252/as.2020.1801

Abstract

A many identities group (MI-group, for short) is an algebraic structure which is generalized a monoid with cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In other words, an MI-group is an algebraic structure generalizing the group concept, except most of the elements have no  inverse element. The concept of a topological MI-group, as a preliminary study,  in the paper '' Topological MI-group: Initial study'' was introduced by M. Hol\v capek and N. \v Skorupov' a, and we  have given a more comprehensive study of this concept in our two recent papers.  This article is a continuation of the effort to develop the theory of topological MI-groups and is focused on the study of  separation axioms and  the isomorphism theorems for topological MI-groups.  Moreover, some conditions under which a MI-subgroup is closed  will be investigated, and finally, the existence of nonnegative  invariant measures on the locally compact MI-groups are introduced.

Keywords

#### References

[1] H. Bagheri and S.M.S. Modarres, Internal Topology on MI-groups, Alg. Struc. Appl. Vol. 5 No. 2 (2018) 55-78.
[2] H. Bagheri and S.M.S. Modarres, A note on topological MI-groups, preprint.
[3] A.M. Bica, Categories and algebraic structures for real fuzzy numbers, Pure Math. Appl. 13(12) (2003) 63-67.
[4] D. Fechete, I. Fechete. Quotient algebraic structure on the set of fuzzy numbers, Kybernetika 51 (2) (2015) 255-257.
[5] E. Hewitt, and K.A. Ross, Abstract harmonic analysis, Vol. 1. Springer Verlag, Berlin, 1963.
[6] M. Holcapek, M. Wrublova, M. Bacovsky, Quotient MI-groups, Fuzzy sets and syst. 283 (2016) 1-25.
[7] M. Holcapek, M.  Stepnicka, MI-algebras: A new framework for arithmetics of (extensional) fuzzy numbers, Fuzzy Sets Syst. 257 (2014) 102-131 .
[8] M. Holcapek. On generalized quotient MI-groups. Fuzzy Sets Systems 326 (2017), 3-23.
[9] M. Holcapek, M.  Stepnicka. Arithmetics of extensional fuzzy numbers - part I: introduction. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, (2012) 1517-1524 .
[10] M. Holcapek, M.  Stepnicka. Arithmetics of extensional fuzzy numbers - part II: Algebraic framework. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, (2012) 1525-1532.
[11] M. Holcapek, N.  Skorupova. Topological MI-groups: Initial Study. In: Medina J., Ojeda-Aciego M., Verdegay J., Per lieva I., Bouchon-Meunier B., Yager R. (eds) Information Processing and Management of
Uncertainty in Knowledge-Based Systems. Applications. IPMU 2018. Communications in Computer and Information Science, vol 855. Springer, Cham, 603-615.
[12] M. Mares, Computation over Fuzzy Quantities. CRC Press, Boca Raton (1994).
[13] S. Markov, On the algebra of intervals and convex bodies. J. Univ. Comput. Sci. 4(1) (1998) 34-47.
[14] S. Markov, S. On the algebraic properties of convex bodies and some applications. J. Convex Anal. 7(1) (2000) 129-166.
[15] A. Mukherjea, N. A. Tserpes, Invariant measures and the converse of Haar; s theorem on semitopological
semigroups, Paci c J. of Math., Vol. 44, No. 1 (1973), 251-262 .
[16] J. R. Munkres, Topology; A First Course, Prentice-Hall, Inc., Englewood Cli s, New Jersey.
[17] R. Rigelhof, Invariant measures on locally compact semigroups, Proc. Amer. Math. Soc., Vol. 28, (1971), 173-175.
[18] D. Qiu, C. Lu, W. Zhang, Y. Lan, Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mares equivalence relation. Fuzzy Sets and Systems 245 (2014), 63-82.