I was reading this article which explains proof of compactness theorem using los's theorem. I read until the definitions given for reduced products, ultraproducts, ultrapower in page 15. Now in page 19 they provided that they extend the definition of ultraproducts over sets and rather define them for structures(They did it for models).
My question:-
$(1)$- I didn't understand first of all what an atomic formula is, I request to give an example if possible. So i was thinking if it's same as equality relation in general. For example, let $x_1,x_2$ be two variables (All variables are terms). When I say those two variables are same then does it mean that $x_1=x_2$??
Is it correct way to think?
$(2)$- I even didn't understand the definition of ultraproduct they mentioned for structures? I even request an explanation of it in possibly deep way.