I'm working on a generalization of Boolean algebra in a way that it can encompass 'series' of true/false values. My motivation is to introduce logic gates which are mapping from $\{0, 1\}^n$ to $\{0, 1\}^n$; for example, generalized OR gate would map [1 1 0 1] and [1 0 0 0] to [1 1 0 1]. Long term goal would be to create a customizable network (build from those logic gates) which extracts intervals of interest.
I read about Allen's interval algebra, but as far as I understand, it's purpose is temporal reasoning and not the computation itself. Is there any already established mathematical framework which introduces structures/notions of this 'generalization' of Boolean algebra?
Thank you in advance! :)