MATH.18:5.4 - Recover an operation using a chosen origin

On the integers, one account supplies the ternary operation t(x,y,z)=x-y+z and a marked origin 0. It recovers addition as t(x,0,y) and negation as t(0,x,0). Conversely, addition and negation recover t. Substitution verifies both returns for every integer input.

Now remove the marked origin. Choosing any integer c gives an addition x+_c y=t(x,c,y)=x-c+y, identity c and inverse 2c-x. Combining these operations again gives the same t(x,y,z). The ternary operation alone therefore permits several choices of origin and corresponding binary operations.

For example, c=0 makes the sum of 2 and 3 equal 5; c=1 makes it equal 4 under +_1. Both choices recover the original ternary operation. A question using only t can continue without selecting an origin; recovering the former addition needs its marked origin again. This distinguishes a usable one-way interpretation from a return that silently supplies extra structure.