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Provides a categorical approach to quantales and applications. Develops the theory of modules on unital quantales. Includes exercises and bibliographical notes.

In applications, for example in health care, many-valuedness modelled using quantales plays an important role. The paper presents variations of the three chain modules C-3 over unitalization of the three chain quantale (C-3 is the smallest possible quantale to model many-valuedness), thus, variations of right actions are given. From application point of view, it is then possible to choose suitable modules when modelling, for example, the causalities between disease, intervention and functioning. The effect of drug interaction in presence of multiple diseases, and as affecting functioning, adds to this complexity. Health care communities and professionals comply with a range classifications and terminologies, also including scales to qualify strength or hierarchies of evidence, in the sense of Evidence-Based Medicine (EBM), or interaction, or as related to levels of functioning. Such hierarchies adopted in health care are ad hoc as compared to the potentially algebraic and logical structures of terminologies used as part of clinical decision-making. The notion of 'evidence' in EBM would benefit from a logical enrichment. In this paper we show how these hierarchies canonically derive as actions where transitions appear as levels in hierarchies of evidence. In this sense, the Logic-Based Medicine (LBM) view is complementary to EBM, the latter indeed very well known and established, whereas LBM is yet to be recognized more broadly. We will also see how three-valuedness related to health conditions, rather than two-valuedness, is the generator of many-valuedness related to strength of evidence.

In the past there have been made various attempts to define the spectrum of a non-commutative C*-algebra. But all these definitions have certain drawbacks - e.g. C.J. Mulvey's definition does not coincide with the standard definition of the spectrum in the commutative case. The aim of this paper is to give an alternative definition of the spectrum which does not suffer under this deficit - i.e. coincides with the standard situation in the commutative setting. For this purpose we recall some properties of balanced and bisymmetric quantales, introduce a definition of the spectrum of a C*-algebra working for the general case and develop subsequently its topological representation.

Let Sup be the category of complete lattices and join preserving maps. The aim of this paper is to show that the theory of complete many-valued lattices exists. For this purpose we recall the concept of many-valued preordered sets and show that the category of many-valued join-complete lattices is isomorphic to the category of right modules in Sup - a theorem which goes back to I. Stubbe in a more general context given by quantaloid enriched categories. Finally, we will explain the presented theory by some natural examples, and do not hesitate to apply the theory to a simple stochastic linear programming problem.

In this paper we propose a logical enrichment DSM so that design structure is described in a logical language rather than as an unstructured relation. We further show how DSM potentially can embrace languages and notations that support modelling of information and process.

We present three paradigms for non-classical substitution in a many-sorted context. Such an exposition has previously been demonstrated in the unsorted case but its extension is far from trivial. The first paradigm, classical many-sorted substitution taking variables to terms, is traditionally presented in a rather informal and "verbal" manner but we find that a strict categorical formulation is necessary to pave the way for non-classical extensions. The second paradigm provides substitution of variables for many-valued sets of terms and relies heavily on functors and monads over the category of indexed sets. Finally, in the third paradigm, we establish full non-classical substitution of many-valued sets of variables by many-valued sets of terms. The third paradigm has the category of many-valued indexed sets as its underlying category. These paradigms ensures transparency of the underlying categories and also makes a clear distinction between set-theoretic operation in the meta language and operations on sets and many-valued sets as found within respective underlying categories.