Leometry and groupls

As a youngster I have discovered an un-obvious symmetry for a family of certain (generalized) tic-tac-toe games. This immediately has induced me into defining an attractive topic (its formulation is so natural that it had to exist forever–I am not claiming any credit, this topic belongs naturally to the public domain): given a family of sets  \(F\),  study the group of all permutations of  \(\bigcup F\)  onto itself, which preserve  \(F\).  I mentioned this topic many times but I never studied it, which is a pity. Of course this topic generalizes finite geometries or some infinite too (e.g. affine geometry and projective geometry, with their groups–affine and projective–which preserve the family of straight lines).

Now I will formulate a more special topic in the same spirit. (First let me publish what I have already written here–I’d like to make sure that mathjax, or is it jaxmath, works, or more specifically the inline \(\LaTeX\)).

(It does not. Let me check and eventually try to extend the respective java script).

(My fault!!! MathJax does not support the in-line code between the single dollar signs, but it does so between slash-parenthesis 2-term sequences. Now the \(\LaTeX\) code should show up. It is also possible to enable the single dollar option. BTW, I wanted this and the previous comment written in a small font–how do I do it? the small-tag pair doesn’t work.)


Let  \((L\ G)\)  be an ordered pair consisting of a set  \(L\),  called line, and a group  \(G\)  of permutations (i.e. bijections onto itself) of  \(L\).  Then let  \(G_\bullet\)  be  \(G\)  together with all constant functions  \(c:L\rightarrow L\).

The geometries (or rather leometries) described below are interesting (nontrivial) only when  \(|L|\ge 2\).

A leometry (i.e. line geometry) over  \((L\ G)\)  is an ordered triple  \(\Gamma\ :=\ (X\ P\ \Lambda)\),  where:

  • \(X\)  is a set called  space,  its elements are called points;
  • \(P\)  is a set of functions  \(\pi : X\rightarrow L\),  called functionals or projections;
  • \(\Lambda\)  as a the set of all injections  \(\lambda: L\rightarrow X\),  called parametrizations or orbits,  and such that

    \[\forall_{\lambda\in\Lambda}\forall_{\pi\in P}\quad \pi\circ\lambda\in G_\bullet\]

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Now we define a \(\Gamma\)-line as an image  \(L’ := \lambda(L)\)  for arbitrary  \(\lambda\in\Lambda\);  and let  \(\mathcal L\)  be the set of all \(\Gamma\)-lines.

Given a leometry  \(\Gamma\),  the goal is to describe the groupl  \(G_\Gamma\)  of  \(\Gamma\),  which is defined as the group of all bijections  \(T:X\rightarrow X\)  such that:

\[\forall_{L’\in\mathcal L}\quad T(L’) \in \mathcal L\]