Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the kind of groups studied by Galois and others. Then the axioms were abstracted from the concrete example and it was proved that any abstract group was isomorphic to a concrete group.

I'm trying to repeat this process with different concrete structures on sets, as $\mathcal P(X)$ and $\text{End}(X)$, just for fun. Both of these examples seems more complicated, however. $\text{Aut}(X)$ rather obvious boils down to an associative composition $\circ$, an identity $e$ and an inverse $f^{-1}$ for all functions $f\in\text{Aut}(X)$.

But how to deal with $\text{End}(X)$? Again there is an associative composition $\circ$ and an identity $e$, which point towards monoids. But here it seems to exist hidden structures: unlike for bijections (and surjections), there is a non-trivial function $\text{Im}:\text{End}(X)\to \mathcal P(X)$, that makes algebraically closed subsets (including $e$) of $\text{End}(X)$ to somewhat special monoids. And the question is, how to formulate the axioms for this concrete structure?

One idea would be to study a monoid action on sets

$$\text{End}(X)\times\mathcal P(X)\to\mathcal P(X),\;(f,A)\mapsto f(A)$$

but that ends up in finding axioms for the $\mathcal P(X)$-structure with it's own more or less hidden structures.

Is there a way to abstract axioms for $\text{End}(X)$ without dealing with $\mathcal P(X)$?

I'm interested in canonically pre-structured sets.

The simplest one is $X\times X$ with the structures $\displaystyle X\times X\overset {p_1,p_2}{\longrightarrow} X,\;p_i(x_1,x_2)=x_i$. Obviously there are no more structures. For the sets of all bijections $Bij(X)$ or surjections $Sur(X)$ on a set $X$ it is rather obvious that there is one canonical structure: $Bij(X)\times Bij(X)\to Bij(X)$ etc.

Now suppose the example $N(X)$ has a binary $N(X)\times N(X)\to N(X)$ structure and a topological structure, canonically. If you're only interested in one of those structures you might describe the axioms for that structure and try to show that all the abstract objects are isomorphic with a concrete subset of $N(X)$. But it might also be of interest to find the axioms for both structures and their interaction.

An abstract monoid doesn't have an extra structure corresponding to $\text{Im}:\text{End}(X)\to \mathcal P(X)$, but the functions in End$(X)$ do have it.

  • $\begingroup$ Just to clarify: End and Aut are relative to unstructured sets? i.e. $\text{End}(X)$ is the set of all functions from $X$ to $X$ and $\text{Aut}(X)$ is the set of all invertible functions from $X$ to $X$, right? Also, when you speak of special monoids induced by a submonoid of $\text{End}(X)$, is the result a submonoid of $\text{End}(\mathcal{P}(X))$? I think it's clear from context, but I am uncertain! $\endgroup$ – Hugh Denoncourt Jan 13 '15 at 1:03
  • $\begingroup$ @HughDenoncourt: Yes, End$(X)$ and Aut$(X)$ is just sets of functions, but I really don't know about a submonoid of End$(\mathcal P(X))$..? I just think it is a complication. $\endgroup$ – Lehs Jan 13 '15 at 5:34
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    $\begingroup$ Given any monoid $M$, there is an embedding of $M$ into $\text{End}(M)$. The result and proof is similar to Cayley's theorem for groups. In other words, abstract monoids are concrete monoids. The monoid axioms suffice to study algebraically closed subsets of $\text{End}(X)$. The paragraph regarding the image function Im led me to believe you are looking to characterize monoids arising from submonoids (ie algebraically closed subsets with e) of $\text{End}(M)$ via Im. I guess I'm unclear what concrete structure you seek axioms for in that paragraph. $\endgroup$ – Hugh Denoncourt Jan 13 '15 at 6:04
  • $\begingroup$ Related, but I don't think this contains your $\mathcal{P}(X)$ question: math.stackexchange.com/questions/218353/… $\endgroup$ – Hugh Denoncourt Jan 13 '15 at 6:08
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    $\begingroup$ A given closed subset of $\text{End}(X)$ is necessarily a monoid, but of course, it could satisfy additional axioms. (e.g. $\text{Aut}(X) \subset \text{End}(X)$ and is closed under $\circ$ and $e$ and satisfies the group axioms.) But if you add an axiom that isn't a consequence of the monoid axioms, there will exist a closed subset of some $\text{End}(X)$ that doesn't satisfy that axiom. I understood your question as asking whether the class of subsets of $\text{End}(X)$'s of a particular special form have additional axioms and its own "Cayley representation theorem". This is possible. $\endgroup$ – Hugh Denoncourt Jan 13 '15 at 18:14

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