In this paper we investigate generalized topologies generated by a subbase of preorder relators and consider its application in the concept of the complement. We introduce the notion of principal generalized topologies obtained from the new type of open sets and study some of their important properties.

A topology τ on a fixed nonempty set X is said to satisfy the weakly continuous representation property if every weakly continuous not necessarily total preorder on the topological space (X, τ) admits a continuous order preserving function. Such a property generalizes the well known continuous representation property of a topology τ on a set X (according to which every continuous total preorder...

I will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then pr...

The eXtensible Markup Language (XML) is becoming a new standard for information representation and exchange over the Internet. How to index XML data for efficient query processing and XML transformation is an important subject in the XML community. In this paper, based on extended preorder indexing method, we add path information as part of the index. It is shown that the number of path joins c...

We examine several extensions and variants of Plotkin’s language PCF, including non-deterministic and probabilistic choice constructs. For each, we give an operational and a denotational semantics, and compare them. In each case, we show soundness and computational adequacy: the two semantics compute the same values at ground types. Beyond this, we establish full abstraction (the observational ...

We design a new variation on the picalculus, πcost, in which the use of channels or resources must be paid for. Processes operate relative to a cost environment, and communications can only happen if principals have provided sufficient funds for the channels associated with the communications. We define a bisimulation-based behavioural preorder in which two processes are related if, intuitively...

In this paper, we present a general way of giving denotational semantics to a class of languages equipped with an operational semantics that ts the GSOS format of Bloom, Istrail and Meyer. The canonical model used for this purpose will be Abramsky's domain of synchronization trees, and the denotational semantics automatically generated by our methods will be guaranteed to be fully abstract with...

Term rewriting is an appealing technique for performing program analysis and program transformation. Tree (term) traversal is frequently used but is not supported by standard term rewriting. In this paper, many-sorted rst-order term rewriting is extended with automatic tree traversal by adding two primitive tree traversal strategies and complementing them with three types of traversals. These s...

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory...