📝Algebraic Subtyping
Theory books recommendations:
- Introduction to Lattices and Order by Davey and Priestley
- Categories for the Working Mathematician by Mac Lane
- Stone Spaces by Johnstone (mix of order and category theories)
- Galois connections and fixed point calculus by Backhouse (Kleene algebra)
theory:
- preorder ≤ on a set A is reflexive transitive binary relation.
- ≡ is a kernel of preorder such that a ≡ b iff a ≤ b and b ≤ a
- partial order is a preorder satisfying antisymmetry (if a ≤ b and b ≤ a, then a = b)
- partial order is preorder which kernel is equivalence (=)
- poset is a set equipped with partial order
- if S ≤ A then upper bound of S is
a
of A such that s ≤ a for any s in S - least upper bound or join is upper bound b of S such that b ≤ a for any a = upper bound of S
- meet is greatest lower bound
- ⊔S - join
- ⊔∅ = ⊥
- ⊓S - meet
- ⊓∅ = ⊤
- lattice is a poset where every finite subset has a meet and join (has finite joins and meets)
- join-semilattice is a poset that has finite joins
- meet-semilattice is a poset that has finite meets
- complete lattice is a poset that has arbitrary joins and meets (not necessary of finite subsets)
- ^ same for complete semilattices