Elements of generalized ultrametric domain theory

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultra-metric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations that combines and extends the standard order-theoretic (Smyth and Plotkin, 1982) and metric (America and Rutten, 1989) approaches.

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