Birkhoffs Theorem from a geometric perspective: A simple example

‎From Hilberts theorem of zeroes‎, ‎and from Noethers ideal theory‎, ‎Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original exles of algebraic geometry‎. ‎I will describe a simple exle that illustrates some of the aspects of this relationship‎. The dualization from algebra to geometry in the basic Grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method‎, ‎known as Isbell conjugacy.