Using the well-known categorical notion of ‘functor’, one may define the concept of datatype (algebra) without being forced to introduce a signature (that is, names and typings for the individual sorts (types) and operations involved). This has proved to be advantageous for those theory developments where one is not interested in the syntactic appearance of an algebra.

The categorical notion of ‘transformer’ developed in this paper allows the same approach to laws: without using signatures one can define the concept of law for datatypes (lawful algebras), and investigate the equational specification of datatypes in a syntax-free manner. A transformer is a special kind of functor and also a natural transformation on the level of dialgebras. Transformers are quite expressive, satisfy several closure properties, and are related to naturality and Wadler'' Theorems For Free Theorem. In fact, any colimit is an initial lawful algebra.

In this paper we try to shed some light on the similarities and differences in the different approaches denning the notions of implementation and implementation correctness. For obvious reasons, we do not discuss all existing approaches individually. Instead, a formal framework is introduced in order to discuss the most important ones. Additionally, we discuss some issues, which in our opinion are often misunderstood, concerning transitivity of implementation correctness and its role in the software development process. In particular, on the one hand, we show that for reasonable notions of implementation, it is almost impossible to prove transitivity of implementation correctness at the specification level. On the other hand, we show that this is not really important if the programming language satisfies the properties of horizontal and vertical composition.

This paper considers two main aspects of the lower power locale PL(X): first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of X with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from o(X) to Sub(1), and a new proof of a theorem of Jibladze relating them to Ω-nuclei.

Non-strict don't care functions, whose foremost representative is the ubiquitous if_then_else, play an essential role in computer science. As far as the semantics is concerned, they can be modelled by their totalizations with the appropriate use of elements representing undefinedness, as D. Scott has shown in his denotational approach. The situation is not so straightforward when we consider non-strict functions in the context of an algebraic framework; this point is discussed in the last section, where we explore the relationship between non-strict don't care and total algebras. The central part of this paper, after presenting the basic properties of the category of non-strict algebras, is an investigation of conditional algebraic specifications. It is shown that non-strict conditional specifications are equivalent to disjunctive specifications, and necessary and sufficient conditions for the existence of initial models are given. Since non-strict don't care specifications generalize both the total and the partial case, it is shown how the results for initiality can be obtained as specializations.