Abstract

bases were introduced in [Smyth, 1977] where they are called "R-structures". Examples of abstract bases are concrete bases of continuous domains, of course, where the relation OE is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation OE on an abstract basis need neither Domain Theory 25 be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An ideal A in a basis, for example, has the property (following from directedness) that for every x 2 A there is another element y 2 A with x OE y. In po...

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