Mark Gritter (markgritter) wrote,
Mark Gritter

Lego lattices

I commented in the previous entry that the collection of Lego constructions doesn't form a semilattice, because while lower and upper bounds exist, they are not unique.

What I was thinking about was a definition where all "Lego constructions" are connected components. Constructions are assumed to have rotational and mirror symmetry around the vertical axis, but the presence of the tabs eliminates horizontal symmetries. Ordering is by "piece inclusion": X < Y if X contains only pieces in Y. With this definition, a simple counterexample to uniqueness of bounds is these two pieces:

While both a single red brick and a single yellow brick are lower bounds, they are unrelated to each other. So there is no greatest lower bound. (Including both yellow and red bricks makes the construction unconnected, and thus invalid in the definition I was using.)

Color is not necessary for this example, you could put two different structures that don't share any pieces in the same locations to make the same idea work with monochrome bricks.

Is there a way to make a semi-lattice, though? What if we restricted ourselves to Lego constructions on an infinite base plate? Then neither the lone yellow brick nor the lone red brick would be valid constructions. Is there an example which shows that "simply connected Lego constructions including an infinite base plate" still does not have greatest lower bounds?
Tags: category theory, geek, lego, mathematics
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