By definition, a map between topological spaces is continuous if preimages of open sets are open or, equivalently, preimages of closed sets are closed. Can one get similar characterizations for localic maps between locales? The characterization of localic maps in terms related to continuity may be a bit delicate: complements of closed sublocales have to be formed in the Heyting algebra of all sublocales and not set-theoretically in the Boolean algebra of all subsets of the space, as in classical topology.

In the talk, we will address this problem in the more general setting of implicative semilattices, that is, meet-semilattices with top element in which the unary meet operations λ_a = a ∧ (-) have adjoints α_a = a → (-). Locales and frames are just the complete implicative semilattices, frame homomorphisms are nothing but the residuated semilattice homomorphisms preserving the top element. This setting provides generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) point-free setting.

It turns out that the "continuous morphisms" between implicative semilattices (the so-called localizations) are characterized by the following closure-theoretical continuity properties: the preimage of the zero ideal (the least basic closed set) is zero, the preimage of any basic closed set is basic closed, and its complement in the lattice of α-ideals is contained in the preimage of the complement -- a triviality in the counterpart of set-theoretical complements, but an unavoidable additional condition in the lattice-theoretical setting. In the complete case of locales, the prefix "basic" is avoidable, since basic closed sets form then a closure system.

This is a joint work with Marcel Erné and Ales Pultr.