what does it mean to extend an assignment? For a constraint satisfaction problem, what does it mean for  an assignment x to extend an assignment  a?  

Sorry if this is super trivial,  I did not find an answer

e.g here:  No Small Linear Program Approximates Vertex Cover Within a Factor 2 − $ \epsilon $  pg 16 x does not extend \alpha
It means that you have a partial solution, and you extend that partial solution by assigning at least one more variable to some value, thus producing either a new partial solution or a solution where every variable is assigned a value.

Put differently, if $x$ extends an assignment $a$, it holds that $x$ does precisely the same assignments as $a$ and assigns any number of other variables in addition.