Get started
Work

Complete Lattices as Frameworks for Interval and General Type-2 Fuzzy Inference Systems

Year: 2025
Type: article
Abstract: The secondary membership functions (MFs) of type-2 (T2) fuzzy sets (FSs) are generally assumed to be convex, normal, and upper semicontinuous. Although not explicitly stated, this fact can be detected by taking a look at the$\alpha$-cuts of their join and meet in vertical-slice format. These two operations induce$[0,1]_{\mathcal {F}}$, i.e., the complete lattice of the convex normal upper semicontinuous FSs on$[0, 1]$together with the so-called convolution order. In this article, we deduce the$\alpha$-cut representations of suprema and infima over finite and infinite sets that occur in the inference mechanisms of general type-2 (GT2) fuzzy inference systems (FISs). We continue by developing a complete, albeit not distributive, lattice structure for the class of all convex upper semicontinuous FSs on$[0, 1]$, denoted${\mathcal {F}}_{CU}$, which permits designing T2 FISs involving T2 FSs whose secondary MFs are not necessarily normal. Using our formulae of the$\alpha$-cuts of suprema and infima in$[0,1]_{\mathcal {F}}$and${\mathcal {F}}_{CU}$, whose elements we describe using so-called anti-dilations, we prove that$[0,1]_{\mathcal {F}}$is a closed sublattice of${\mathcal {F}}_{CU}$. We also expand on the lattice-theoretical relations between$[0,1]_{\mathcal {F}}$and the complete lattices of the convex (strictly) normal FSs. In addition, we show that$[0,1]_{\mathcal {F}}$and${\mathcal {F}}_{CU}$contain complete sublattices of (extended) interval type-2 (IT2) FSs. With these results at hand, it becomes easy to perform the transitions FISs under the minimum t-norm$\to$(closed) IT2 FISs$\to$GT2 FISs currently used in practice$\to$GT2 FISs having not necessarily normal secondary MFs.
DOI: 10.1109/tfuzz.2025.3545295
Language: English
Open Access status: hybrid
Estimated APC paid: $2,645
OpenAlex ID: W4408030080
FWCI:
Cites: 48
Cited by:
Related to: 10