  We introduce a new class of abstract structures, which we call generalized
ultrametric semilattices, and in which the meet operation of the semilattice
coexists with a generalized distance function in a tightly coordinated way. We
prove a constructive fixed-point theorem for strictly contracting functions on
directed-complete generalized ultrametric semilattices, and introduce a
corresponding induction principle. We cite examples of application in the
semantics of logic programming and timed computation, where, until now, the
only tool available has been the non-constructive fixed-point theorem of
Priess-Crampe and Ribenboim for strictly contracting functions on spherically
complete generalized ultrametric semilattices.
