Let Σ be the signature made up from the following symbols.

e: 0 arguments function (constant symbol)

f: 2 arguments function

g: 1 argument function

Variable set Var is made up from x,y,z

Let E be the set of the following equations

f(x,f(y,z)) = f(f(x,y),z)

f(x,e) = x

f(g(x),x) = e

Can someone explain to be me how I can prove that the following equations are non-derivable from E

f(e,x) = x

f(x,g(x)) = e

  • 1
    $\begingroup$ Give a model where the equations of $E$ hold, but where the target equations don’t. $\endgroup$ – Yuval Filmus May 22 '19 at 7:05
  • $\begingroup$ How do I come up with such a model? $\endgroup$ – A. Othmane May 22 '19 at 7:06
  • $\begingroup$ Try to construct a model whose universe is very small, say two elements. $\endgroup$ – Yuval Filmus May 22 '19 at 7:14
  • $\begingroup$ Do you have an example of an operation with identity element on the right but not on the left? $\endgroup$ – A. Othmane May 22 '19 at 7:34
  • 1
    $\begingroup$ It's your exercise. Give it a shot. $\endgroup$ – Yuval Filmus May 22 '19 at 7:54

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