Equivalences

Abstract

In this chapter, we focus on equivalences in nilpotent logical systems. We study three different types of equivalence operators and resolve the paradox that there is no equivalence relation in a non-Boolean setting which fulfills ∀xe(x,x)=1 and e(x, n(x) ) = 0 by aggregating the implication-based equivalence and its dual operator. We will also show that the aggregated equivalence has some nice properties like associativity, threshold transitivity, and T-transitivity. For applications in image processing, we define the overall equivalence of two grey level images and give an important semantic meaning of the aggregated equivalences. Finally, for applications in image processing, we define the overall equivalence of two grey level images and give an important semantic meaning to the aggregated equivalences.