Liang et al. in 2000 defined interval type-2 fuzzy sets (IT2FSs), which constitute a subset of type-2 fuzzy sets. While the membership degrees in the former are functions from [0, 1] to [0, 1] (fuzzy truth values), the membership degrees in IT2FSs only take their values in {0,1}\lbrace {\text{0}},{\text{1}}\rbrace. Although all the initial work on IT2FSs involved convex membership degrees only, in 2015, Bustince et al. began the study on IT2FSs in general, including certain sets with nonconvex membership degrees. However, these are obviously early stages, with a lot of open problems regarding the theoretical structure of IT2FSs. For example, as far as we know, no negation operator has been obtained in this context. Therefore, it seems appropriate to continue with the study started in previous papers, delving deeper into the properties and operations of IT2FSs. Consequently, this work studies the structure of the set of functions from [0, 1] to {0,1}\lbrace {\text{0}},{\text{1}}\rbrace (expanding the set considered by Bustince et al.), from which we have removed the constant function 0\mathbf{0}, to offer a different study to the one carried out by Walker and Walker. More specifically, we consider join and meet operations, partial order derived from each one, and the negation operators in that set. Among other results, we provide new characterizations of join and meet operations and of partial orders on the set of functions from [0, 1] to {0,1} \lbrace {\text{0}},{\text{1}} \rbrace ; we also present the first negation operators on this set.

中文翻译：

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一般区间2型模糊集的补充研究


梁等人。 2000年定义了区间2型模糊集（IT2FSs），它构成2型模糊集的子集。前者的隶属度是从 [0, 1] 到 [0, 1] 的函数（模糊真值），而 IT2FS 中的隶属度仅取 {0,1}\lbrace {\text{0} 中的值。 },{\text{1}}\r大括号。尽管 IT2FS 的所有初始工作仅涉及凸隶属度，但在 2015 年，Bustince 等人。开始了对 IT2FS 的一般研究，包括某些具有非凸隶属度的集合。然而，这些显然还处于早期阶段，关于 IT2FS 的理论结构还有很多悬而未决的问题。例如，据我们所知，在这种情况下还没有获得否定运算符。因此，继续之前论文中开始的研究似乎是合适的，更深入地研究 IT2FS 的属性和操作。因此，这项工作研究了从 [0, 1] 到 {0,1}\lbrace {\text{0}},{\text{1}}\rbrace 的函数集的结构（扩展 Bustince 考虑的集合）等人），我们从中删除了常数函数 0\mathbf{0}，以提供与 Walker 和 Walker 进行的研究不同的研究。更具体地说，我们考虑连接和相遇操作、从每个操作导出的偏序以及该集合中的否定运算符。除其他结果外，我们还提供了从 [0, 1] 到 {0,1} \lbrace {\text{0}},{\text{1} 的函数集的 join 和 meet 操作以及偏序的新特征} \r大括号;我们还提出了该集合上的第一个否定运算符。