A theory of quantale-enriched dcpos and their topologization

There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale is integral and commutative, then -valued domains are -enriched, and every -enriched domain is sober in its Scott -valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of -enriched dcpos and -enriched domains in the general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right subdistributive quasi-magmas on in the sense of the category . It is important to point out that our quasi-magmas on are in tune with the «intersection axiom» of -enriched topologies. When is involutive, each -enriched domain becomes sober in its -enriched Scott topology. This paper also offers a perspective to apply -enriched dcpos to quantale computation.