MV-Partitions and MV-Powers

MV-Partitions and MV-Powers

C. Drossos, P. L. Theodoropoulos
DOI: 10.4018/978-1-4666-4991-0.ch011
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In this chapter, the authors generalize the Boolean partition to semisimple MV-algebras. MV-partitions together with a notion of refinement is tantamount a construction of an MV-power, analogous to Boolean power construction (Mansfield, 1971). Using this new notion we introduce the corresponding theory of MV-powers.
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1. Introduction

In this introduction we should like to touch on some points that will help the readers to comprehend the contents of the Chapter.

1.1 Interpretations and the Change of Point of View

In modern mathematics we usually use Set Theory to express mathematical notions. In Set theory, as such there, is no uncertainty involved. However we usually have the habit to interpret some of the set-theoretic notions, in such a way that we model uncertainty. For example the function 978-1-4666-4991-0.ch011.m01is a well-defined entity in Set Theory, and no uncertainty is connected with such a function. However we may interpret this as a generalized indicator or a membership function of a fuzzy set. This change of point of view leads to Fuzzy Set theory. Similarly a real number978-1-4666-4991-0.ch011.m02, is interpreted in fuzzy set theory as a fuzzy real number978-1-4666-4991-0.ch011.m03. Related to the change of point of view the following Dieudonne’s saying is instructive: “What changes in Mathematics, as in all other Sciences, is the point of view from which results already acquired, are assessed. ”

The change of point of view essentially is connected with interpretations from one model to another. Think, for example, the interpretation of Euclidean terms in Riemannian geometry. To give a technical development of the intuition of interpretations it is a hard work, see e.g. (W. Hodges, 1993, Ch. 5).

Similarly, a functor, e.g. going from the category of Topological Spaces to the Category of Groups, can be construe as a change of point of view, from a topological-geometrical to algebraic.

1.2 Generalized Elements and Generalized Properties

Since functions play a very important role in changing a point of view from a classical set theoretic model to a non-classical one, we should like to comment on the two basic interpretations for the concept of function:

  • Variable or Generalized Elements: Instead of classical elements 978-1-4666-4991-0.ch011.m04we consider variable ones978-1-4666-4991-0.ch011.m05. T is considered as the domain of variation or the stage of definition of the variable element.

  • Generalized Properties: Every indicator function 978-1-4666-4991-0.ch011.m06describes a crisp property978-1-4666-4991-0.ch011.m07. We would like to generalize this to have instead of 2, any set 978-1-4666-4991-0.ch011.m08or 978-1-4666-4991-0.ch011.m09and in general any set V.

Thus a fuzzy set 978-1-4666-4991-0.ch011.m10from this point of view is a generalized property. Any generalized property 978-1-4666-4991-0.ch011.m11 induces on X a partition,

In this way partitions and generalized properties are strongly interconnected.

One general approach to this and similar problems is through Category Theory. We know that there is a “reversing the arrows duality” between subsets and partitions see (Lawvere & Rosebruch, 2003 and Ellerman, 2010). In particular we are interesting in the duality of “generalized elements” and “generalized properties” (Lawvere & Rosebruch, 2003).

We would like to present this reversing the arrows duality in some characteristic examples.

Let first consider a function 978-1-4666-4991-0.ch011.m13 as a diagram in the category of sets, using as an index the graph978-1-4666-4991-0.ch011.m14. Then the graph Gf, is the limit of the diagram 978-1-4666-4991-0.ch011.m15 as depicted in Figure 1.

Figure 1.

Limit of diagram 978-1-4666-4991-0.ch011.m16


In this way we may have a notion of a graph of an arrow in any category with finite limits.

To reveal the connection of Boolean powers with the reversing the arrows duality we have to calculate explicitly the reverse of the function 978-1-4666-4991-0.ch011.m17.

On the domain A we define the equivalence relation:

Let now,
the equivalence classes of the relation978-1-4666-4991-0.ch011.m20. Then it is clear that978-1-4666-4991-0.ch011.m21 constitutes a partition of A. Thus we have the following function:




Thus we get the Boolean power of B with respect to power Boolean algebra978-1-4666-4991-0.ch011.m24. If we express (∗),i.e. 978-1-4666-4991-0.ch011.m25, where 978-1-4666-4991-0.ch011.m26, using the co-graph,


Then the dual of, 978-1-4666-4991-0.ch011.m28is978-1-4666-4991-0.ch011.m29. This is the exact meaning of reversing the arrows in Set. The above meaning of co-graph can be expressed as a co-limit of the diagram,978-1-4666-4991-0.ch011.m30 (see Figure 2)

Figure 2.

Co-limit of diagram978-1-4666-4991-0.ch011.m31


We can repeat the above construction in the categories of probability spaces and in the category of probability algebras as well, giving the essence of the construction of Boolean powers.

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