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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-cuts of their join and meet in vertical-slice format. These two operations induce <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,1]_{\mathcal {F}}$</tex-math></inline-formula>, i.e., the complete lattice of the convex normal upper semicontinuous FSs on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0, 1]$</tex-math></inline-formula> together with the so-called convolution order. In this article, we deduce the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0, 1]$</tex-math></inline-formula>, denoted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {F}}_{CU}$</tex-math></inline-formula>, which permits designing T2 FISs involving T2 FSs whose secondary MFs are not necessarily normal. Using our formulae of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-cuts of suprema and infima in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,1]_{\mathcal {F}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {F}}_{CU}$</tex-math></inline-formula>, whose elements we describe using so-called anti-dilations, we prove that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,1]_{\mathcal {F}}$</tex-math></inline-formula> is a closed sublattice of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {F}}_{CU}$</tex-math></inline-formula>. We also expand on the lattice-theoretical relations between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,1]_{\mathcal {F}}$</tex-math></inline-formula> and the complete lattices of the convex (strictly) normal FSs. In addition, we show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,1]_{\mathcal {F}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {F}}_{CU}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\to$</tex-math></inline-formula> (closed) IT2 FISs <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\to$</tex-math></inline-formula> GT2 FISs currently used in practice <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\to$</tex-math></inline-formula> GT2 FISs having not necessarily normal secondary MFs.