A theory of quantale-enriched dcpos and their topologization

Abstract

There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral and commutative, then Q-valued domains are Q-enriched, and every Q-enriched domain is sober in its Scott Q-valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of Q-enriched dcpos and Q-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 Q in the sense of the category Cat(Q). It is important to point out that our quasi-magmas on Q are in tune with the «intersection axiom» of Q-enriched topologies. When Q is involutive, each Q-enriched domain becomes sober in its Q-enriched Scott topology. This paper also offers a perspective to apply Q-enriched dcpos to quantale computation.