✍ Scribed by K.J. Nüßler
No coin nor oath required. For personal study only.
In this paper we i n vestigate the Plotkin powerdomain under order-theoretical aspects. We answer a problem of G. Plotkin whether any bi nite domain can be embedded (with embedding-projection pairs) into the Plotkin powerdomain of a Scott-domain. Here we obtain counter-examples. There is a 9-element domain which cannot even be embedded into the Plotkin powerdomain of any bi nite domain. In the case of L-domains, we nd that there is no mub-complete domain whose Plotkin powerdomain is universal for the class of all L-domains. However, any nite domain can be \weak mub-embedded" into the Plotkin powerdomain of a nite Scott-domain, where a weak mub-embedding is an order-embedding f which preserves minimality of upper bounds of sets. The class of all Scott-domains as well the class of all L-domains is not closed under the Plotkin powerdomain. We g i v e an order-theoretical characterisation of those Scott-domains and L-domains whose powerdomain is again a Scott-domain or L-domain, respectively. F or Scott-domains we obtain: the powerdomain of those Scott-domains into which the posets M and W (just like the letters) cannot be order-embedded is itself a Scott-domain.
The class of all Scott-domains as well as the class of all L-domains is not closed under the Plotkin powerdomain construction. We obtain the following ordertheoretic characterisation of those Scott-domains and L-domains whose Plotkin powerdomains are again Scott-domains or L-domains, respectively. Theorem 1.4 Let (D ) be a S c ott-domain. Then the following are e quivalent:
(1) P D] is a Scott-domain.
📜 SIMILAR VOLUMES
In this paper we initiate the study of measurements on the probabilistic powerdomain. We show how measurements on an underlying domain naturally extend to its probabilistic powerdomain, so that the kernel of the extension consists of exactly those normalized measures on the kernel of the measurement
The purposes of this paper are (i) to give a new, efficient representation of Smyth powerdomains as 0-inhabited information systems, following the style of Scott; (ii) to introduce, as a consequence of this representation, a generalized resolution method for an arbitrary Smyth powerdomain and show t