# Prooving equations are non derivable in Sigma algebra

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

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