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

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.