In this You Tube video, at 3:39 : https://www.youtube.com/watch?v=enZpq8jvFEs
Monsieur Phi ( understand " Monsieur Philosophie") a philosophy teacher ( and also a logician) gives a visual presentation of Euclid's deductive system of geometry as an ordered set that looks like a lattice.
Is actually a deductive theory a lattice? ( It seems doubtful however, since the axioms of a theory are all minimal elements).
If not, how to caracterize formally the order of a deductive theory?
Are there different possible types of order for deductive theories?
Is it possible for two deductive theories to instantiate two different kinds of order?