The International Symposium on Domain Theory (ISDT) is a conference series intended to be a forum for researchers in domain theory and its applications. Topics include topological and logical aspects of domains; categories of domains and powerdomains; continuous posets and their representations; partial orders, lattice theory and metric spaces; types, process algebra and concurrency; non-classical and partial logics; programming language semantics; applications in computer science and mathematics. This conference series was founded by Yingming Liu, Yixiang Chen, Klaus Keimel, and Guo-Qiang Zhang. All ISDT events have taken place in China. The first ISDT was held in Shanghai, October 17–24, 1999; the second ISDT was held in Chengdu, October 22–26, 2001; the third ISDT occurred in Xi'an, China, May 10–14, 2004; the fourth ISDT was held in Changsha, June 2–6, 2006; and the fifth ISDT took place in Shanghai, September 11–14, 2009.

In this paper, we study the relationship between ordering structures on semirings and semiring-induced valuation algebras. We show that a semiring-induced valuation algebra is a complete (resp. continuous) lattice if and only if the semiring is complete (resp. continuous) lattice with respect to the reverse order relation on semirings. Furthermore, a semiring-induced information algebra is compact, if the dual of the semiring is an algebraic lattice.

In this paper, we introduce the notion of consistent F-augmented contexts by adding a special family of finite subsets into the structure of a formal context, which essentially establishes the basis of the representation of general algebraic domains. In particular, we investigate the association rule systems which are derived from the consistent F-augmented contexts and propose the notion of formal association rule systems. By the notion of antecedent connections, we obtain the equivalence between the category of formal association rule systems and that of algebraic domains, which demonstrates that the proposed notion of formal association rule systems provides a concrete approach to representing algebraic domains.

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to Escardó which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never previously for projective structures. Using some examples, we advertise the versatility of this approach – in particular, highlighting its advantage over existing methods on characterisation of projectives, which is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the coalgebra arising from the comonad.

A topological space X is called well-filtered if for any filtered family $\mathcal{F}$ of compact saturated sets and an open set U, ∩ $\mathcal{F}$ ⊆ U implies F ⊆ U for some F ∈ $\mathcal{F}$. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every UK-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each T1 space X we consider a dcpo D(X) whose maximal point space is homeomorphic to X and prove that X is well-filtered if and only if D(X) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.

This paper studies the fuzzy Scott topology on dcpos with a *-continuous semigroup (L, *) as the truth value table. It is shown that the fuzzy Scott topological space on a continuous dcpo is an ιL-sober space. The fuzzy Scott topology is completely distributive iff L is completely distributive and the underlying dcpo is continuous. For (L, *) being an integral quantale, semantics of L-possibility of computations is studied by means of a duality.

In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

In 1978, G. Plotkin noticed that $\mathbb{T}$ω, the cartesian product of ω copies of the three element flat domain of Booleans, is a universal domain, where ‘universal’ means that the retracts of $\mathbb{T}$ω for 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 a least upper bound. Since $\mathbb{T}$ω is also an ωDI-domain (an important structure in stable domain theory), the following problem arises naturally: is there a cartesian closed category C of domains with stable functions such that $\mathbb{T}$ω, or a related structure, is universal in C for Berry’s stable semantics? The aim of this paper is to answer this question. We first investigate the properties of stable retracts. We introduce a new class of domains called conditionally complete DI-domains (CCDI-domain for short) and show that, (1) $\mathbb{T}$ω is an ωCCDI-domain and the category of CCDI-domains (resp. ωCCDI-domains) with stable functions is cartesian closed; (2) [$\mathbb{T}$ω →st$\mathbb{T}$ω] is a stable universal domain in the sense that every ωCCDI-domain is a stable retract of [$\mathbb{T}$ω → st$\mathbb{T}$ω], where [$\mathbb{T}$ω → st$\mathbb{T}$ω] is the stable function space of $\mathbb{T}$ω; (3) in particular, [$\mathbb{T}$ω → st$\mathbb{T}$ω] is not a stable retract of $\mathbb{T}$ω and hence $\mathbb{T}$ω is not universal for Berry’s stable semantics. We remark that this paper is a completion and correction of our earlier report in the Proceedings of the 6th International Symposium on Domain Theory and Its Applications (ISDT2013).