it is in the very first sentence of the proof:
Let n be the maximal length of a strictly decreasing chain of non-zero
principal right ideals of S.
I'm having trouble with a proof I've found on the web. Could you please help me? It's Lemma 2.0.1 here (Ash's Theorem, Finite Nilpotent Semigroups, and One-Dimensional Tiling Semigroups, D. B. McAlister).
The theorem reads:
Lemma 2.0.1 Let S be a finite nil semigroup. Then S is nilpotent. The nilpotent index of S is the maximal length of a strictly decreasing chain of principal non-zero ideals of S.
A nil semigroup is a semigroup with zero in which every element is nilpotent. A nilpotent semigroup a semigroup with zero for which there exists n>0 such that any product of n+1 elements of S is 0.
I understand the first part of the proof, where it is proved that a finite nil semigroup S is nilpotent, and that the nilpotent index is at most the maximal length of a strictly decreasing chain of principal non-zero ideals of S. What I don't understand is when the author says
"On the other hand, there is a strictly decreasing chain of non-zero principal right ideals of S."
I have no idea what this follows from.
the a_{i} (in the first part) are completely arbitrary elements. he uses a chain built from n+1 of them to show that 2 must be equal (since we can only have a chain of length n that is STRICTLY decreasing).
on the other hand, from a given strictly decreasing chain of length n (assumed to exist from the outset, and there may be several such), he constructs a word of length n, showing (from the first part) that the nilpotency is both less than n+1, and at least n, and therefore n.
i assume that by 0 he means "absorbing element" (i.e. z: az = z for all a in S), like for example the 0 map of the semigroup End(V), for a vector space V.
Yes, that's what 0 means in semigroup theory, except we need az=za=z. Otherwise such an element is called a one-sided zero. Thanks again! I was completely unable to understand that because I thought he used the same symbols to mean the same things. Since the first were completely arbitrary, I couldn't understand where he took the inclusions from. I'm just this slow on the uptake.
By the way, how do you get the subscript to work outside the tex tags?