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Three Lessons on the Topological and Algebraic Hidden Core of Rough Set Theory

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Algebraic Methods in General Rough Sets

Part of the book series: Trends in Mathematics ((TM))

Abstract

In what follows the reader will find an exposition of the basic, albeit not elementary, connections between Rough Set Theory and relation algebra, topology and algebraic logic.

Many algebraic aspects of Rough Set Theory, are known nowadays. Other are less known, although they are important, for instance because they unveil the “epistemological meaning” of some “unexplained” mathematical features of well-known algebraic structures.

We shall wrap everything in a simple exposition, illustrated by many examples, where just a few basic notions are required. Some new results will help the connection of the topics taken into account.

Important features in Rough Set Theory will be explained by means of notions connecting relation algebra, pre-topological and topological spaces, formal (pre) topological systems, algebraic logic and logic.

Relation algebra provides basic tools for the definition of approximations in general (that is, not confined to particular kind of relations). Indeed, these tools lead to pairs of operators fulfilling Galois adjointness, whose combinations, in turn, provide pre-topological and topological operators, which, in some cases, turn into approximation operators.

Once one has approximation operators, rough sets can be defined. In turn, rough set systems can be made into different logico-algebraic systems, such as Nelson algebras, three-valued Łukasiewicz algebras, Post algebras of order three, Heyting and co-Heyting algebras.

In addition, in the process of approximation, one has to deal with both exact and inexact pieces of information (definable and non-definable sets). Therefore, the concept of local validity comes into picture. It will be extensively discussed because it links the construction of Nelson algebras from Heyting ones with the notions of a Grothendieck topology and a Lawvere-Tierney operator.

As a side effect, we obtain an information-oriented explanation of the above logico-algebraic constructions which usually are given on the basis of pure formal motivations.

The exposition will move from abstract levels (pointless) to concrete levels of analysis.

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Notes

  1. 1.

    Often, a lower adjoint is called “left adjoint” and an upper adjoint is called “right adjoint”. We avoid the terms “right” and “left” because they could make confusion with the position of the arguments of the operations on binary relations.

  2. 2.

    A direct proof of point 6 of Facts 3.1 runs as follows. Let \(x\in \mathcal C_R(X\cap Y)\). Therefore, ∃y, x ∈ R(y) and R(y) ⊆ X ∩ Y . It follows that R(y) ⊆ X and R(y) ⊆ Y , so that \(x\in \mathcal C_R(X)\) and \(x\in \mathcal C_R(Y)\) which amounts to \(x\in \mathcal C_R(X)\cap \mathcal C_R(Y)\). Therefore, \(\mathcal C_R(X\cap Y)\subseteq \mathcal C_R(X)\cap \mathcal C_R(Y)\).

  3. 3.

    In the infinite case there can occur a dual situation: there are not enough points to separate properties. In this case the dual operation of T 0-ification is called spatialisation.

  4. 4.

    If H is finite then it has the least dense element. An infinite Heyting algebra, instead, may lack this element.

  5. 5.

    One obtains a Boolean algebra from a Heyting one by applying another Lawvere-Tierney operator, namely B x(p) = (p⇒x)⇒x. The congruence relation is a ≡ b iff a⇒x = b⇒x. If x = 0, a ≡ b iff ¬a = ¬b. By definition a is a fixed point of B x if (a⇒x)⇒x ≤ a, so that if x = 0, a is a fixed point if ¬¬a ≤ a, hence if ¬¬a = a.

  6. 6.

    Moreover, by means of ⊔ we have a proof that ¬¬ preserves ⇒:

    $$\displaystyle \begin{aligned} \neg\neg(a\Longrightarrow b)&= \neg(\neg\neg a\wedge\neg b) = \neg(\neg\neg a\wedge\neg\neg\neg b)\\ & = \neg\neg(\neg a\vee\neg\neg b)= \neg a\sqcup(\neg\neg b)\\ & = \neg(\neg\neg a)\sqcup \neg\neg b=\neg\neg a\Longrightarrow\neg\neg b \end{aligned} $$

    The last equation is legal because it is calculated in the Boolean algebra \(\mathcal F_{J^\delta }(\mathbf {H})\) of the regular elements of H.

  7. 7.

    The first proof was presented in [28]. In that paper there is a misprint: ¬(¬B 2 ∪ A 2) instead of (¬B 2 ∪ A 2).

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Author information

Authors and Affiliations

  1. International Rough Set Society, Rome, Italy

    Piero Pagliani

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  1. Piero Pagliani
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Editors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, Kolkata, India

    A. Mani

  2. Department of Informatics, University of Milano-Bicocca, Milano, Italy

    Gianpiero Cattaneo

  3. Fujian Normal University, College of Mathematics and Informatics, Fuzhou, China

    Ivo Düntsch

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Pagliani, P. (2018). Three Lessons on the Topological and Algebraic Hidden Core of Rough Set Theory. In: Mani, A., Cattaneo, G., Düntsch, I. (eds) Algebraic Methods in General Rough Sets. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01162-8_4

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