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Has anyone ever heard of (can you think of) of a group structure on $\mathbb{P}$, the set of all polyominoes? Ideally the monomino would be our identity element, I'd say. (Thanks to Justin Lanier for the great question.)

$\mathbb{P}$ is countable, so of course we can jam the structure of the integers or rationals on top of it, or any countable group for that matter. I want something that relates to the geometric properties of the polyominoes.

I've built operations that have inverses for every element but are not associative or well-defined. I'd love to hear that there's something out there that I just can't find, but I'd also love to hear your thoughts on the problem. Thanks!

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What is a polyomino? –  Metin Y. Sep 30 '13 at 13:21
    
@metina an edge-connected set of edge-aligned squares with equivalence defined by geometrical transformations. Or do you have some other definition in mind that is useful here? –  Jan Dvorak Sep 30 '13 at 13:25
    
one square = monomino; two squares = domino; ... ; five squares = pentomino; ... . There are twelve double-sided (invariant under mirror reversal) pentominoes (with a standard set of letter mnemonics). –  Jan Dvorak Sep 30 '13 at 13:31
    
That looks good, @JanDvorak. If it's useful for the problem, we could drop the rotations and call those distinct. It's also common to disallow reflections. –  Paul Salomon Sep 30 '13 at 13:32
    
@paul that's not the standard procedure for polyominoes but you are free to discuss the case - it could yield some interesting results. –  Jan Dvorak Sep 30 '13 at 13:40
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