On the concept of fuzzy order II: Antisymmetry

Abstract

In the second part of our paper, we explore antisymmetry of fuzzy orders. We provide a unifying definition of antisymmetry, which generalizes three existing variants of antisymmetry examined in the literature, along with the corresponding generalized definition of fuzzy order. We prove that all the particular instances of the generalized definition, which include the three basic ones, are mutually equivalent. We also examine distinctive properties of the three basic notions of fuzzy order.

Keywords:

• Order
• fuzzy logic
• fuzzy equality
• antisymmetry

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In fact, Blanchard in general defines the notion of a fuzzy order on a fuzzy set defined on the universe U. The notion we describe corresponds to the case of a fuzzy order on a set, i.e. when the fuzzy set on U is identified with U.

2 With respect to this notion of fuzzy order, Zhang and Fan (2005) cite an earlier paper by L. Fan, Q.-Y. Zhang, W.-Y. Xiang, and C.-Y. Zheng, “An L-fuzzy approach to quantitative domain (I) (generalized ordered set valued in frame and adjunction theory)”, Fuzzy Systems Math. 14 (2000), 6–7, written in Chinese, which we were not able to obtain.

3 A frame, or a complete Heyting algebra, is a complete lattice satisfying $a\wedge \left(\underset{j}{\bigvee }{b}_j\right)=\underset{j}{\bigvee }(a\wedge b_j)$. That is, a frame may be regarded as a complete residuated lattice in which ⊗ coincides with the infimum $\wedge$.

4 As is easily seen, further variations of the claims of the two theorems may be formulated. For instance, a variation of Theorem 1 may be proved for fuzzy orders according to Definition 1 as well as for fuzzy orders according to Definition 2.

Additional information

Funding

This work was supported by grants IGA_PrF_2022_018 and IGA_PrF_2023_026 of Palacký University Olomouc.