Algebraic properties and topological properties of the quotient space of fuzzy numbers based on Mareš equivalence relation

In this paper, we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by Mareš: every fuzzy number has only one Mareš core and equivalent fuzzy numbers have the same Mareš core; in addition, equivalence classes of fuzzy numbers can be only expressed as the sum of its Mareš core and the set of symmetric fuzzy numbers, which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group. Based on these results, we introduce a new concept of convergence under which the quotient space is complete. As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there would be more differentiable fuzzy functions than what is found in the literature.