Equivalence With Algebraic Lattices
There is a well-known equivalence between the category of join-semilattices with zero with -homomorphisms and the category of algebraic lattices with compactness-preserving complete join-homomorphisms, as follows. With a join-semilattice with zero, we associate its ideal lattice . With a -homomorphism of -semilattices, we associate the map, that with any ideal of associates the ideal of generated by . This defines a functor . Conversely, with every algebraic lattice we associate the -semilattice of all compact elements of, and with every compactness-preserving complete join-homomorphism between algebraic lattices we associate the restriction . This defines a functor . The pair defines a category equivalence between and .
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