Tω as a Stable Universal Domain

In the seventies, G. Plotkin noticed that Tω, the cartesian product of ω copies of the 3 elements flat domain of Boolean, is a universal domain, where “universal” means that the retracts of Tω in Scott's continuous semantics are exactly all the ωCC-domains, which with Scott continuous functions form a cartesian closed category. As usual “ω” is for “countably based”, and here “CC” is for “conditionally complete”, which essentially means that any subset which is pairwise bounded has an upper bound. Since Tω is also an ωDI-domain (an important structure in the stable domain theory), a problem arises naturally: Is Tω a universal domain for Berry's stable semantics? The aim of this paper is to answer this question. We investigate the properties of stable retracts and introduce a new domain named a conditionally complete DI-domain (a CCDI-domain for short). We show that, (1) a dcpo is a stable retract of Tω if and only if it is an ωCCDI-domain; (2) the category of ωCCDI-domain (resp. CCDI-domains) with stable functions is cartesian closed. So, the problem above has an affirmative answer.