I am looking for how far algebra on types in the manner of "set theory as an algebra" ¹ is possible.

So for example in set theory one can compute for sets S, T
S∪T, S∩T, S-T etc

Is something similar possible for types?

Say I have
data Primary = Red|Green|Blue
data Othercolors = Violet|Indigo|Yellow|Orange

I want something like
Rainbow = Primary ∪ Othercolors

Equivalently if Rainbow and Primary had been defined, how to get/compute
Rainbow - Primary?

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I thought the first-class types in Idris would be a good bet to try out at least a trivial prototype.

Seems not...

So asking here.

Clearly and obviously one can use haskell to implement any language.

My question is what/which are the introspective libraries/features of modern haskell that make this easy and lightweight.

Thanks

Rusi

¹ Yeah the term 'type algebra' may be taken in the sense of algebraic data types

Cant think of a better one