special relativity - Associativity of Relativistic Oblique Velocity Addition

I've encountered some information in the Wikipedia page on Lorentz transformation (https://en.m.wikipedia.org/wiki/Lorentz_transformation) that I am having difficulty reconciling with other information that I found on the page (https://en.m.wikipedia.org/wiki/Wigner_rotation) on Wigner rotation. In the Lorentz Transformation page, it says that the oblique (non-collinear) Lorentz transformation is given by a matrix $$\begin{pmatrix}\gamma&-\gamma\beta n_x&-\gamma\beta n_y&-\gamma\beta n_z\\-\gamma\beta n_x&1+(\gamma-1)n_x^2&(...Read more

example of non-associativity in the physical universe?

In a recent article[1], John Baez is quoted as making a nice point about how non-commutativity is common in the world around us, whereas non-associativity is not: “[...] while it’s very easy to imagine noncommutative situations—putting on shoes then socks is different from socks then shoes—it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parenth...Read more

Associativity 0f Łukasiewicz T-norm Fuzzy Set

I'm looking at proving that the Łukasiewicz T-norm operator, or bounded product T-norm, is a T-norm, but I'm stuck on associativity. The operator is defined as:$xTy = max[0,x+y-1]$Trying to show associativity: $(aTb)Tc =$$max[0,max[0,a+b-1]+c-1] =$ $max[0,max[c-1, a+b+c-2]] =$$max[0,max[c-a,b+c-1]+a-1] =$$max[0,a+max[c-a,b+c-1]-1] =$At this point, I need to show that this is equivalent to...$=max[0,a+max[0,b+c-1]-1]=aT(bTc)/$But I can't figure out how to resolve $c-a$ to zero. Hopefully someone can help!...Read more