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While implementing an embryo of computational algebra on my blog I ran into a rather special monoid and I wonder if it's studied before.

After implementing a very simple concept of dynamic sets I used it to study permutations and permutation groups. The permutation groups represents finite groups fine since all finite groups are isomorphic to a group of permutations, but when computing quotient groups the elements are no longer permutations but sets of permutations.

One of my ideas to generalize the concept is to consider all groups as quotient groups. Any permutation group $G$ could be considered as the group $G/(e)$, subgroups $N\subset G$ as subgroups $N/(e)\subset G/(e)$ and quotient group $G/(e)/N/(e)$ as $G/N$. Then all group elements are sets of permutations and the composition of elements is composition of sets of permutations.

But only some sets of permutations generates groups. The unitary sets and the cosets build up with normal subgroups are among them. Generally a set of permutations doesn't generate a cyclic group but are merely an element in the monoid $M(S_n)=\mathcal P(S_n)-\{\emptyset\}$ with $2^{n!}-1$ elements. The set of the identity in $S_n$ is the identity in $M(S_n)$.

Is this monoid studied? It would be nice if it has properties that can be used in computational algebra for making smart code.

Lehs
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  • I found it a bit hard to follow precisely what this monoid is. Also, they are called quotients, not quotes. – Tobias Kildetoft Jan 25 '16 at 20:05
  • Thanks! The monoid is the power set of $S_n$, the symmetry group, and two subsets of permutations are composed in the obvious way. – Lehs Jan 25 '16 at 20:10
  • What kind of computations are you planning to carry out? – Rob Arthan Jan 25 '16 at 20:15
  • @RobArthan. On my blog? I'm just playing around to see what's possible for me. – Lehs Jan 25 '16 at 20:30
  • So clearly one can define this for any group, or even any monoid or semigroup. I have a feeling I have seen it referred to before, but it may have just been in relation to a question along the same lines. – Tobias Kildetoft Jan 25 '16 at 20:40
  • OK, have fun! The monoid of subsets of a group under "aggregate mutiplication" is used implicitly all the time in group theory texts, but I don't l know of any computational results about it, but I am no expert. I should point out that there is a huge literature on computational algebra and computational group theory (but, again, I am now expert $\ddot{\frown}$). Tou may want to look into this literature if very efficient representations become important to. I think the leading computational group theory system is GAP. – Rob Arthan Jan 25 '16 at 20:42
  • @TobiasKildetoft, yes you're right, but it's concrete groups that I could possibly be able to implement. I should maybe try to search on the more general case. – Lehs Jan 25 '16 at 20:45
  • @RobArthan: so it's used then. I will see if I find something. If I want to make progress in efficiency I might try to find some elementary text book on computational algebra. So GAP is the best? – Lehs Jan 25 '16 at 20:52
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    I am not sure how GAP compares to MAGMA for example, but GAP has the huge advantage of being open source. It is also implemented in SAGE, so one can do things with it through there. – Tobias Kildetoft Jan 25 '16 at 20:54
  • @Lehs Is the obvious way to compose subsets as follows: Given $S, T \subseteq \mathcal{P}(S_n)$, $ST = {u | \exists s\in S, t\in T, st = u}$? So, if $S = {x}$ and $T = {x^{-1}}$, does $ST = {e}$? – mhum Jan 25 '16 at 20:58
  • @Lehs: yes, the notation $XY = {xy : x \in X, y \in Y}$ is very common. E.g., see the statement of the second isomorphism theorem here. – Rob Arthan Jan 25 '16 at 21:00
  • @mhum: your formula for $ST$ is the right one (and works for any group $G$, not just $S_n$). It makes $\mathcal{P}(G)$ into an associative monoid with ${e}$ as the identity. The submonoid comprising singleton sets is isomorphic to $G$: ${x}{x^{-1}}= {e}$, and ${x}{y} = {xy}$, but if $S$ has two or more elements it has no inverse. – Rob Arthan Jan 25 '16 at 21:04
  • @RobArthan. The cosets in the quotient groups has inverses and identity that isn't inverses or indentities in the monoid. – Lehs Jan 25 '16 at 21:19
  • @Lehs: sure: if you restrict the monoid operation to particular subsets of $\mathcal{P}(G)$, inverses and identities may appear again. I was talking about the full monoid $\mathcal{P}(G)$, which is what you were asking about (with $G = S_n$). Note that if you restricts to the cosets of a normal subgroup, you are looking at a subsemigroup but not a submonoid (the multiplication is the same, but the identity elements are different). – Rob Arthan Jan 25 '16 at 21:24
  • @RobArthan: Thanks, I think the most is sorted out now in the comments. I'll look for the monoid in text books of group theory. – Lehs Jan 25 '16 at 21:32
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    Here's a link to an MSE question that might help http://math.stackexchange.com/questions/101045/what-can-we-learn-about-a-group-by-studying-its-monoid-of-subsets. – Rob Arthan Jan 25 '16 at 21:37

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