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**12 July 2017**

**08:00AM - 10:00AM**

##### Room: **Catalana**

##### Chairs: **Didier Dubois, Stephen E. Rodabaugh and Austin Melton**

## Fuzziness and the Mathematics of Many-Valuedness

**Abstract - ** Let Sup be the category of complete lattices and join preserving maps. The aim of this talk 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 2006 in a more general context given by quantaloid enriched categories. Finally, the presented theory will be explained by some natural examples.

**Abstract - ** In the past there have been made various attempts to define the spectrum of a non-commutative C-star-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 our talk 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-star-algebra working for the general case and develop subsequently its topological representation.

**Abstract - ** In this paper we define the Smarandache hoopalgebras and Q-Smarandache filters, we obtain some related results. After that, by considering the notions of these filters we determine relationships between filters in hoop-algebras and QSmarandache filters in hoop-algebras. Finally, we introduce the concept of Smarandache 2-structure and Smarandache 2-filter on hoop-algebras.

**Abstract - ** 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 over unitalization of the three chain quintile (three chain 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. Effects of drug interactions 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) 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 logic structures of terminology infused reasoning. In this paper we show how these hierarchies canonically derive as actions where transitions appear as levels in hierarchies of evidence. We will also see how three-valuedness related to health conditions, rather than two-valuedness, is the generator many- valuedness related to strength of evidence.

**Abstract - ** In this paper fuzzy (tied) relational systems are considered which are the objects of semicategories whose morphisms constitute a general variable-basis approach to fuzzy Galois connections and conjugated pairs. Useful applications to some kinds of algebraic structures are outlined.

**Abstract - ** The purpose of this paper is to make a case for the value of many-valued mathematics, often called fuzzy mathematics. We believe there may be a difference between many-valued mathematics and fuzziness, as used by those who work with fuzzy logic and fuzzy set theory and applications thereof. We think that most, if not all, fuzzy mathematics is many-valued. However, for this paper, the difference between many-valued mathematics and fuzzy mathematics, if a difference exists, is not important. We are. in this paper, content to show that many-valued mathematics can contribute to mathematics. We do understand that for those mathematicians who feel that many-valued mathematics does not have a place in mathematics this paper will not cause them to embrace many-valued mathematics, but we would like for some of them to consider that many-valued mathematics might be able to contribute to mathematics. In this paper, we give an example of a mathematical construction which was created and defined in part to help computer scientists understand and be able to use topological ideas and concepts in their work as computer scientists. Thus, one would think that this construction, called topological systems, would be topological (as defined later). However, it seems that topological systems are clearly not topological. Thus, an interesting question is can topological systems be made topological, or said more mathematically, can topological systems be embedded into something which is topological.