Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent _L_-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an _L_-domain, and every _L_-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that (...) the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of _L_-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting. (shrink)

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains with Scott continuous (...) functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, _L_-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions. (shrink)