On the geometry of join and meet calculations for general type-2 fuzzy sets

The union and intersection of general type-2 fuzzy sets (T2 FSs) are fundamental computations for such FSs. In the past, algorithms were developed for the union and intersection computations using vertical-slice and horizontal-slice representations of T2 FSs. The vertical-slice representation of a T2 FS traces its origins back to Zadeh [32] and requires computing the join or meet, whereas the horizontal-slice representation of a T2 FS is very recent, traces its origins to Liu [14], and requires computing the join and meet only for interval T2 FSs that are raised to level alpha, for which closed-form formulas are available [15]. In this paper, by studying the join and meet geometrically for general T2 FSs, we show that for many situations closed-form formulas exist for them. We also show that the formulas for computing the union and intersection of general T2 FSs that were derived from the horizontal-slice representation of a T2 FS [19] can also be obtained directly from geometrical interpretations of formulas that were derived by Karnik and Mendel [12] for the vertical-slice representation of a T2 FS.