Peculiar properties of the $p$-linear decomposition of $p$-linear functions in terms of the shift-composition operation

We analyze the shift-composition operation on discrete functions which occurs under homomorphisms of finite shift registers. We prove that for a prime $p$, in the class of all functions that are linear in the extreme variables, the notions of reducibility and $p$-linear reducibility coincide for $p$-linear functions. Furthermore, we show that a linear function irreducible in the class of all linear functions has no $p$-linear divisors that are bijective in the rightmost variable, and in some cases, has no $p$-linear divisors at all.