# What is $V : S \to V$?

I came across a problem where I don't know what field to consult. The problem is simple to explain:

I have a geometric object $S$, and a finite set of maps $V$ from the points in this object to $V$.

Intuitively, any $v \in V$ partitions $S$, and chosing one of these partitions yields a new such partitioning.

What kind of thing is this and/or what could I google to get more information on it or similar structures? (I am particularly interested in iterated applications, given an initial element of $V$ (partitioning), to a given sequence of points.)

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where did you come across this problem? – bgins Feb 2 '12 at 8:59
I'm working in logic and this seemed a succinct way of describing an aspect of the semantics I'm investigating (tentatively). – tap Feb 2 '12 at 9:05

This seems like a fairly general concept. For one, it seems to be a particular generalization of finite symmetry groups. Below is some food for thought.

Note that $V^{(k)}=\{v^k|v\in V\}\subset V$ for $k\in\mathbb{Z}^+$.

Note that we could also view each $s\in S$ as defining an action $\sigma(s)$ on $V$ by $v\rightarrow v(s)$. In other words, $(\sigma(s))(v)=v(s)$. For each $s$, let us call $\sigma(s):V\rightarrow V$ the induced action of $s$ on $V$ and let us call $\sigma:S\rightarrow V^V$ the "complementary" map.

Since composition of maps is associative, $V$ is a semigroup under map composition. If there is a unique element $e \in V$ so that $e(S)=\{e\}$ (i.e., a unique simultaneous fixed point of each "complementary" map $s:V\rightarrow V$), then $V$ is a monoid. Note, however, that composition of maps is not commutative. For example, we have no guarantee that right-inverses and left-inverses of each $v \in V$ exist (in $V$) or that they are the same. Of course, if this is the case, then $V$ is a group as well. But in general, $V$ will be a semigroup and we need to be aware of concepts like pseudoinverse, idempotents, Green's relations and absorbing elements before we try to think of $V$ as the more familiar group. Each $V^{(k)}$ above will be a sub-object of the respective type. In case $V$ is in fact a group (which is not true in general), it would help to have some background on the symmetric group of permutations.

If there exists an $n>1$ so that $v^n=v$ for all $v$ or, more generally, so that $V^n=V$ these would be interesting cases to study. The latter case (when the map $v\rightarrow v^n$ is onto or surjective) might imply that $V$ is a group with unique identity, particularly for $n=|V|$, but you might also need to assume a limit on the number of idempotents. Whether left or right inverses are always well-defined seems to also hinge on the number of distinct idempotents.

If left or right inverses are well-defined and there is at most one idempotent, I think we can extend $V$ to include an identity and left or right inverses of each element. Each stepwise extension would still be finite. Whether the recursive extension (like a closure) is finite is another question.

$V=S$ would be another interesting case to study, but might reduce or specialize to the study of symmetry groups, the symmetric group, and group actions. If $V$ were allowed to be infinite, elliptic curves would be another interesting example.

Intuitively, I'm thinking of the (finite sets) $V$ as "views" or perspectives or projections of the (possibly infinite sets) $S$, and as the "complementary" map $\sigma$ as an important key to decomposing $V$. If $S$ is itself already a group, then this "complementary" map could define a group action of $S$ on $V$, providing a rich source of examples.

Perhaps there are also fruitful connections with category theory.

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Thank you for your elaborate answer. I will certainly look into the relfections you've offered. – tap Feb 2 '12 at 13:28