Necessary and sufficient conditions that $\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_M\rangle$ generates $\mathscr{P}_n.$

Question

Throughout I use cycle notation and write maps $m:X\to Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$).

Let $\zeta=(12\dots n)$.

This question is inspired by the following questions:

• Necessary and Sufficient conditions to generate $S_n$

• Showing the full transformation semigroup $\mathscr{T}_n=\langle\zeta, \tau, \pi\rangle$.

• Showing $\mathscr{P}_n=\langle\zeta, \tau, \pi, \xi\rangle$.

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

Definition 2: For $i\ne j$ in $N$, let $\lvert\lvert i\, j \rvert\rvert$ denote the map $\phi\in\mathscr{T}_n$ for which $$i\phi =j,\quad x\phi=x\quad (x\ne i).$$

Definition 3: The partial transformation semigroup $\mathscr{P}_n$ is the set of all partial maps from $\{1, 2, \dots , n\}$ to itself, together with composition of partial transformations.

Definition 4: For each subset $M$ of $N=\{1, 2, \dots , n\}$, define $\xi_{M}$ by the identity $1_{N\setminus M}$.

Let me interject by saying I'm not sure if this question makes much sense, so if that's the case, I'm sorry.

Question: Are there necessary and sufficient conditions on $1\le i\le j\le n$, $k, \ell\in N$, and $M\subseteq N$ such that $$\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_M\rangle$$ generates $\mathscr{P}_n$?

Thoughts:

I don't have much to offer besides pointing to the lemmas of the questions that inspired this question.

This is not something I think I could solve myself.

The question arose out of curiosity.

I did a module in semigroup theory back in, oh, 2014/2015, and I read about a third of Howie's "Fundamentals of Semigroup Theory" (before starting a PhD in combinatorial group theory). I also did a module in formal languages & automata circa 2012/2013; plenty of monoids were studied then; and my Master's dissertation was on inverse semigroups.

Please help :)

Answer 1

A complete solution to your question can be found in Chapter 3 of the book

O. Ganyushkin and V. Mazorchuk, Classical finite transformation semigroups. An introduction. Algebra and Applications, 9. Springer-Verlag London, Ltd., London, 2009. xii+314 pp. ISBN: 978-1-84800-280-7

Here are the relevant results. Let $\mathcal{S}_n$ be the symmetric group of $\{1, \ldots, n\}$, let $\mathcal{T}_n$ be the full symmetric semigroup of all transformations of $\{1, \ldots, n\}$ and let $\mathcal{PT}_n$ be the semigroup of all partial transformations of $\{1, \ldots, n\}$.

Lemma 1. Each generating system of $\mathcal{PT}_n$ must contain a generating system of $\mathcal{S}_n$. If $A$ is an irreducible generating system of $\mathcal{PT}_n$, then $A \cap \mathcal{S}_n$ is an irreducible generating system of $\mathcal{S}_n$.

Lemma 2. Each generating system of $\mathcal{PT}_n$ must contain at least one element of rank $(n − 1)$.

Theorem 3. Let $A$ be a generating system of $\mathcal{T}_n$. Then this system is irreducible if and only if $A = A_1 \cup \{\alpha\}$, where $A_1$ is an irreducible generating system of $\mathcal{S}_n$ and $rank(\alpha) = n − 1$.

Theorem 4. Let $A$ be a generating system of $\mathcal{PT}_n$. Then this system is irreducible if and only if $A = A_1 \cup \{\alpha, \beta\}$, where $A_1$ is an irreducible generating system of $\mathcal{S}_n$, $\alpha$ is a total transformation of rank $n − 1$ and $\beta$ is a partial permutation of rank $n − 1$.