The Galois Connection between Operations and Relations

sets of relations are denoted by Rel (<κ) A and Rel (<∞) A , respectively. For an operation f ∈ Op A and a relation ρ ∈ Rel (μ) A we say that f preserves ρ, or ρ is compatible with f , and write f ⊥ ρ, if f(ρ, ρ, . . .) ⊆ ρ. Equivalently, f preserves ρ iff ρ is a subalgebra of the algebra 〈A; f〉 iff f if a homomorphism 〈A; ρ〉 → 〈A; ρ〉 of relational structures. For any pair κ, λ where each one of κ, λ is either a cardinal or ∞, compatibility of operations and relations defines a Galois