Dual of the graphic beta move implies some Reidemeister moves
For the extended beta move see the dedicated tag. See also the tutorial “Introduction to graphic lambda calculus“.
The extended beta move is still in beta version. In this post I want to explore some consequences of the dual of the graphic beta move.
I proved earlier the the extended beta move is equivalent with the pair (graphic beta move, dual of the graphic beta move), let me recall this pair in the following picture.
Here are some particular applications of the dual of the graphic beta move. The first is this:
But this is equivalent with the emergent algebra move R1a (via a CO-COMM move). Likewise, another application of of the dual of the graphic beta move is this:
which is the same as the emergent algebra move R2a.
The third application is this one:
and it was discussed in the post “Second thoughts on the dual of the extended beta move“, go there to see if this move may be interpreted as a pruning move (or an elimination of loops).
Finally, there is also this:
which was not mentioned before. It suggests that
behaves like the generic point in algebraic geometry.
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Forgive my lack of knowledge of the names of the objects. The same self-loop but reversed and with one arrow pointed in, the one labeled with a Lambda, would be the dual of the one that you describe as “behaves like the generic point in algebraic geometry”
See the picture posted here: http://www.facebook.com/photo.php?fbid=574102569282387&set=a.574102635949047.147284.100000479464385&type=1&relevant_count=1&ref=nf
I don’t know why, but again I cannot access your link. You might just mail it to me and I shall edit your comment.
Now I understood! Yes, the dual is the “I” combinator, the one at left in this picture.