New representations of algebraic domains and algebraic L-domains via closure systems

Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.