We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.

我们研究隐含半格之间的剩余同态（r-态射）及其伴随，即所谓的局部化（或l-态射），因为这些对象可能被表征为一元相遇操作具有伴随的半格。自从离开了。右伴随图是剩余的对应图。残差映射（具有主要下降和扰乱的原像的性质），人们不仅可以将 l-态射视为无点框架中的抽象连续映射（在完整的情况下很熟悉），而且还可以将它们表征为混凝土闭合理论的连续性。这些概念适用于语言环境（框架、完整的 Heyting 格），并将空间之间的连续和开放映射推广到代数（不一定是完整的）无点设置。