Internal Topology on MI-groups

Document Type : Research Paper

Authors

1 Department of Mathematics, Yazd University, Yazd, Iran

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

Abstract

An MI-group is an algebraic structure based on a generalization of the concept of a monoid that satisfies the cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In this paper, a topology is defined on an MI-group $G$ under which $G$ is  a topological MI-group. Then we will identify open, discrete and compact MI-subgroups. The connected components of the elements of $G$ and connected MI-groups are also identified. Some features of the maximal MI-subgroups and ideals of a topological MI-group are investigated as well. Finally, some theorems about automatic continuity will be introduced.

Keywords

References

[1] Michal Holcapek, Michaela Wrublova, Martin Bacovsky, Quotient MI-groups, Fuzzy sets and syst. 283
(2016) 1-25
[2] E. Hewitt, and K.A. Ross, Abstract harmonic analysis, Vol. 1. Springer Verlag, Berlin, 1963.
[3] James R. Munkres, Topology; A First Course, Prentice-Hall, Inc., Englewood Cli s, New Jersey.
[4] Michal Holcapek, M. Stepnicka, MI-algebras: A new framework for arithmetics of (extensional) fuzzy
numbers, Fuzzy Sets Syst. 257 (2014) 102-131 .
[5] M. Holcapek. On generalized quotient MI-groups. Fuzzy Sets Systems 326 (2017), 3-23.
[6] A.M. Bica. Categories and algebraic structures for real fuzzy numbers. Pure Math. Appl. 13(12), 6367
(2003).
[7] D. Fechete, I. Fechete. Quotient algebraic structure on the set of fuzzy numbers, Kybernetika 51 (2) (2015) 255-257.
[8] M. Holcapek, M. Stepnicka. Arithmetics of extensional fuzzy numbers - part I: introduction. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, pp. 15171524 (2012).
[9] M. Holcapek, M. Stepnicka. Arithmetics of extensional fuzzy numbers - part II: Algebraic framework. In:
Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, pp. 15251532 (2012).
[10] M. Holcapek, N. Skorupova. Topological MI-groups: Initial Study. In: Medina J., Ojeda-Aciego M., Verde-
gay 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.
[11] Mares, M. : Computation over Fuzzy Quantities. CRC Press, Boca Raton (1994).
[12] Markov, S. : On the algebra of intervals and convex bodies. J. Univ. Comput. Sci. 4(1), 3447 (1998).
[13] Markov, S. : On the algebraic properties of convex bodies and some applications. J. Convex Anal. 7(1),
129166 (2000).
[14] 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.
Algebraic Structures and Their Applications
Volume 5, Issue 2
November 2018
Pages 55-78

Files

Share

How to cite

Statistics