Abbreviation: Loop
A \emph{loop} is a structure A=⟨A,⋅,∖,/,e⟩ of type ⟨2,2,2,0⟩ such that
(y/x)x=y, x(x∖y)=y
(xy)/y=x, x∖(xy)=y
e is an identity for ⋅: xe=x, ex=x
Remark:
Let A and B be loops. A morphism from A to B is a function h:A→B that is a homomorphism:
h(xy)=h(x)h(y), h(x∖y)=h(x)∖h(y), h(x/y)=h(x)/h(y), h(e)=e
Example 1:
f(1)=1f(2)=1f(3)=1f(4)=2f(5)=6f(6)=109f(7)=23746f(8)=106228849f(9)=9365022303540f(10)=20890436195945769617f(11)=1478157455158044452849321016