Ytukyg

I'm sorry if this is a duplicate in any way. Throughout I use cycle notation and write maps $m:Xto Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$).

This is Exercise 1.7 of Howie's Fundamentals of Semigroup Theory.

The Details:

Definition 1: Let $S$ be a semigroup and $A$ be a subset of $S$. The subsemigroup of $S$ generated by $A$ is given by the intersection of all the subsemigroups of $S$ containing $A$, denoted $langle Arangle$.

Definition 2: The full transformation semigroup $mathscr{T}_n$ on the set $N={1, dots,n}$ is given by all maps $alpha:Nto N$, together with composition of transformations.

Definition 3: For $ine j$ in $N$, let $lvertlvert i, j rvertrvert$ denote the map $phiinmathscr{T}_n$ for which $$iphi =j,quad xphi=xquad (xne i).$$

Let $pi=lvertlvert 1, 2 rvertrvert$.

Lemma 1: The following hold: $$begin{align}
(1, i)circpicirc (1, i)&=lvertlvert i, 2 rvertrvertquad (ige 3), \
(2, j)circpicirc (2, j)&=lvertlvert 1, j rvertrvertquad (ige 3), \
(i, j)circlvertlvert i, j rvertrvertcirc (i, j)&=lvertlvert j, i rvertrvertquad (i, jge 1, ine j).
end{align}$$

Proof: Just plug & chug.$square$

Lemma 2: Let $varphiinmathscr{T}_n$ with $lvertoperatorname{im}varphirvert=rle n-1$. Let $ine j$ such that $ivarphi=jvarphi$ and let $zin Nsetminusoperatorname{im}varphi.$ Then $$varphi=lvertlvert i, j rvertrvertcirchat{varphi},$$ where $$ihat{varphi}=z,quad khat{varphi}=kvarphiquad (kne i).$$

Proof: This is again just a matter of plug & chug: the maps agree on $N$.$square$

The Question:

Let's get a possible typo out of the way first.

A subquestion: Does $$(1, i)circ(2, j)circlvertlvert i, j rvertrvertcirc (2, j)circ (1, i)=lvertlvert i, j rvertrvert$$ for $i,jge 3, ine j$?

Perhaps I'm being stupid but I can't get the LHS to agree with the RHS on $1$ (as $1mapsto imapsto i mapsto jmapsto 2mapsto 2$ but I need $1lvertlvert i, j rvertrvert=1$). The correct version of this result is considered as part of Lemma 1.

Let $tau=(1, 2), zeta=(1, 2, dots , n)$.

Deduce (from Lemma 1 and Lemma 2) that $mathscr{T}_n=langlezeta, tau, pirangle$.

My Attempt:

I'm not sure where to start. I did the previous question of Howie's book (showing the symmetric group $mathscr{S}_n=langletau, zetarangle$) without any trouble. That might be of some use here. I can see how Lemma 1 might be made to fit $zeta, tau$ and $pi$, making elements of $mathscr{T}_n$.

Please help :)

As for the subquestion: that is definitely a typo, the correct equality is:

$$(1, i)circ(2, j)circlvertlvert 1, 2 rvertrvertcirc (2, j)circ (1, i)=lvertlvert i, j rvertrvert$$

when $i,jnot = 1,2$, $inot=j$.

As for the main question: Let $f_0in T_n$ be arbitrary. Then either $f_0in
S_n$ or $f_0in T_nsetminus S_n$. In the first case, $f_0in langle (1,2),
(1,2cdots n)rangle$ and you're done. In the second case, $f_0in T_nsetminus
S_n$ and so, by Lemma 2, we can write $$f_0=||i_0,j_0||,f_1$$ for some $i_0,
j_0$ and $f_1$ satisfying $operatorname{rank}(f_1)=operatorname{rank}(f_0)+1$. If $operatorname{rank}(f_0)=n-k$
for some $k$, then repeatedly applying Lemma 2 you get
$$begin{align}
f_0&=lvertlvert i_0,j_0rvertrvert,f_1 \
&=lvertlvert i_0,j_0rvertrvertcdotlvertlvert i_{1},j_{1}rvertrvert,f_2 \
&=dots \
&=
lvertlvert i_0,j_0rvertrvertcdotlvertlvert i_{1},j_{1}rvertrvertcdot dots cdot lvertlvert i_{k-1},j_{k-1}rvertrvert,f_k,
end{align}$$
where $f_kin
S_n$ (since $operatorname{rank}(f_k)=n$). It follows that $f_0in langle lvertlvert 1,2rvertrvert, (1,2),
(1,2cdots n)rangle$.