How math deals with state or "algebra of sequential logic"

Question

Boolean algebra is well-studied and you could easily find a lot of introductory materials on different operators, properties, how eveything is nicely related to the algebra of sets, and so on.
Coming from the EE background, I also know that you can construct any circuit using NAND/NOR gates, because of their nice functional completeness properties. The same goes for stateful circuits, as you could build a simple stateful RS-Latch with 2 NAND/NOR gates, by applying them to each other.
This is where it gets interesting - does that mean that the state arises from the recursion? i.e.

 q = nand(r, nand(s, q))

Unfortunately, I couldn't find anything like "algebra of sequential logic" or any notion on how mathematics seems to deal with state (or, better say: time?!). Most likely, my search keywords are compeltely off.
Could you help me out? I would like to know how mathematicians deal with sequential circuits - any introductory book would be great.

Answer 1

The math involved is that of Semigroups and its application to "sequential circuits" is named "algebraic theory of machines".

Read just to start:

Krohn, Rhodes, Arbib - Algebraic Theory of Machines, Languages and Semigroups

Answer 2

I don't know the answer, but I'd like to register a little bit of doubt regarding the position that semigroups/automata provide the most direct account of such phenomena. I'm not saying they aren't related, but I don't think knowing the basic definitions of that branch of knowledge directly answers your question. Anyway, I recommend reading up about traced monoidal categories, which are places where this kind of feedback is allowable. Specifically, you should probably read up on traced cartesian monoidal categories. You may also be interested in this old question of mine at MathOverflow.