Extend completeness and continuity beyond bounded-support fuzzy numbers

Extend the completeness and continuity results established for the uniformly bounded-support fuzzy-number space \(\mathbb{E}^1_{[-M,M]}\) to the entire fuzzy-number space \(\mathbb{E}^1\) and to important subclasses of fuzzy numbers with unbounded support.

Background

The paper develops completeness, cocompleteness, Yoneda completeness, and continuity results for fuzzy numbers whose supports lie within a fixed bounded interval $[-M,M]$. It also shows that the unrestricted space $\mathbb{E}^1$ lacks several stronger completeness properties, including finite conical cocompleteness, finite conical completeness, tensoredness, and cotensoredness.

The authors explicitly identify extending the theory from bounded-support fuzzy numbers to the full space $\mathbb{E}^1$, as well as to fuzzy numbers with unbounded support, as an unresolved direction. Such an extension would determine whether analogous order-theoretic and enriched-domain structures can be obtained without the bounded-support assumption.