Editing Semilattice (section)

==Equivalence with algebraic lattices==

There is a well-known [[Equivalence of categories|equivalence]] between the category <math>\mathcal{S}</math> of join-semilattices with zero with <math>(\vee,0)</math>-homomorphisms and the category <math>\mathcal{A}</math> of [[algebraic lattice]]s with [[compact element|compactness]]-preserving complete join-homomorphisms, as follows. With a join-semilattice <math>S</math> with zero, we associate its ideal lattice <math>\operatorname{Id}\ S</math>. With a <math>(\vee,0)</math>-homomorphism <math>f \colon S \to T</math> of <math>(\vee,0)</math>-semilattices, we associate the map <math>\operatorname{Id}\ f \colon \operatorname{Id}\ S \to \operatorname{Id}\ T</math>, that with any ideal <math>I</math> of <math>S</math> associates the ideal of <math>T</math> generated by <math>f(I)</math>. This defines a functor <math>\operatorname{Id} \colon \mathcal{S} \to \mathcal{A}</math>. Conversely, with every algebraic lattice <math>A</math> we associate the <math>(\vee,0)</math>-semilattice <math>K(A)</math> of all [[compact element]]s of <math>A</math>, and with every compactness-preserving complete join-homomorphism <math>f \colon A \to B</math> between algebraic lattices we associate the restriction <math>K(f) \colon K(A) \to K(B)</math>. This defines a functor <math>K \colon \mathcal{A} \to \mathcal{S}</math>. The pair <math>(\operatorname{Id},K)</math> defines a category equivalence between <math>\mathcal{S}</math> and <math>\mathcal{A}</math>.