Domain Equations Based on Sets with Families of Pre-orders

In this paper we are interested in finding solutions of domain equations based on posets with families of pre-orders. Let (P,⊑) be a poset and let (ω,⩽) be the natural number set. If R=(⊑n)n∈ω is a family of pre-order relations on P, where ⊑0=P×P, such that (i)∀n,m∈ω, m⩽n implies ⊑n⊆⊑m, and (ii)∩n∈ω ⊑n=⊑, then we call (P,⊑) a poset with pre-order family R. We write it R-poset or rpos for short and denote it briefly by (P,⊑;R) [L. Fan, W. Ji and W.L. Wang. The Information Order Approximation and Generalized Chains'Completion, Beijing: Capital Normal University, Preprint, 2005, (in Chinese)]. R-posets are a particular case of quasi-metric spaces (qms) [M.B. Smyth. Quasi uniformities: reconciling domains with metric spaces. Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, APR.8-10,1987. Lecture Notes In Computer Science, Vol. 298, pp 236-253. Springer-Verlag, Berlin, 1988] and generalized ultrametric spaces (gums) [J.J.M.M. Rutten. Elements of Generalized Ultrametric Domain Theory. Technical Report CS-R9507, CWI, Amsterdam, 1995]. R-poset is a ‘nonsymmetric’ version of sfe [L. Monteiro. Semantic domains based on sets with families of equivalences. Electronic Notes in Theoretical Computer Science, 11, pp 1-34, 1998]. We propose a fixed points theorem that can be used for solving domain equations. The paper ends in a final coalgebra theorem.