UPDATE: See also the post “Combinators and stickers“.
Zippers have been introduced here. In particular, the first three zippers are depicted in the following figure.
The combinator has the expression and it satisfies the relation , where means any combination of beta reduction and alpha renaming (in this case is just one beta reduction: ).
In the next figure it is shown that the combinator (figured in green) is just a half of the zipper_1, with an arrow added (figured in blue).
When you open the zipper you get , as it should.
The combinator satisfies . In the next figure the combinator (in green) appears as half of the zipper_2, with one arrow and one termination gate added (in blue).
When you open the zipper you obtain a pair made by and which gets the termination gate on top of it. GLOBAL PRUNING sends B to the trash bin.
Finally, the combinator satisfies . The combinator (in green) appears to be made by half of the zipper_3, with some arrows added and also with a “diamond” added (all in blue). Look well at the “diamond”, it is very much alike the emergent sum gate from this post.