Fields with an overlap between logic and algebra? I was curious if there is any field of study that uses both logic and algebra. To clarify, logic and analysis overlap in areas like descriptive set theory, and there are a lot of logic ideas in areas like topology. But I can't seem to think of any analogous relationships between logic and algebra. I'd appreciate any insight. Thanks 
 A: One such a field is the theory of automata and regular languages. 
I just state here a few results, with links to wikipedia for the definitions, to convince you that algebra and logic really help proving deep theorems in automata theory. There are many more, but unfortunately, most of these research level results are not treated in basic books on automata theory. 
Algebra.
Thm 1 (consequence of Kleene 1956). A language is regular if and only if its syntactic monoid is finite.
Thm 2 (Schützenberger 1965). A language is star-free if and only if its syntactic monoid is finite and aperiodic.
Logic
Thm 3 (Büchi 1960). A language is regular if and only if its definable in monadic second order.
Thm 4 (McNaughton 1971). A language is star-free if and only if it is first order definable.
Thm 5 (Kamp 1968). A language is star-free if and only if it is definable in linear temporal logic.
I would also recommend to read the French entries on Wikipedia for 
Langage rationnel,
Monoïde syntaxique,
Langage sans étoile,
Logique monadique du second ordre,
that are, for some reason, for detailed than their respective English versions.
