Just a little bit of eta goes a long way

L[a,x,b] A[u,x,a] <–eta–> Arrow[u,b] loop

Example: from this post

L[a,x,b] A[b,x,a] <–eta–>

Arrow[b,b] loop <–comb–>

loop loop


L[a,x,b] A[b,x,a] <–beta–>

Arrow[a,a] Arrow[x,x] <–2comb–>

loop loop

Then why not

L[a,x,b] A[u,y,a] <–eta–> Arrow[u,b] Arrow[x,y]

which is exactly alike the FAN-IN

FO[a,x,b] FI[u,y,a] <–FAN-IN–> Arrow[u,b] Arrow[x,y]

Taking this seriously, the beta move should have a hidden companion

FO[a,x,b] FI[b,y,c] <–betahide–> Arrow[y,x] Arrow[a,c]

… which brings us to a symmetrized version of chemlambda which is very close to the interaction nets of Yves Lafont.


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