Approximation of sets based on partial covering

In classic Pawlakian rough set theory the sets used to approximations are equivalence classes which are pairwise disjoint and cover the universe. In this article we give up not only the pairwise disjoint property but also the covering of the universe.

After a historical and philosophical background, we define a general set theoretic approximation framework. First, we reconstruct the rough set theory and partly restate its some well---known facts in the language of this framework.

Next, we present a special approximation scheme. It is based on the partial covering of the universe which is called the base system and denoted by <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>. <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq2.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-definable sets and lower and upper <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq3.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximations are straightforward point---free generalizations of Pawlakian ones. We study such notions as single---layered base systems, <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq4.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-representations of <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq5.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-definable sets, and the exactness of sets. It is a well---known fact that the Pawlakian upper and lower approximations form a Galois connection. We clarify which conditions have to be satisfied by the upper and lower <InlineEquation ID="IEq6"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq6.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximations so that they form a (regular) Galois connection. Excluding the cases when the empty set is the upper <InlineEquation ID="IEq7"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq7.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximation of certain nonempty sets gives rise to a partial upper <InlineEquation ID="IEq8"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq8.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximation map. We also clear up that a partial upper <InlineEquation ID="IEq9"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq9.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximation map and a total lower <InlineEquation ID="IEq10"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-642-36505-8_9_Chapter_TeX2GIF_IEq10.gif" Format="GIF" Rendition="HTML" Type="Linedraw"/></InlineMediaObject><EquationSource Format="TEX">$\mathfrak{B}$</EquationSource></InlineEquation>-approximation map form a partial Galois connection.

In order to demonstrate the effectiveness of our approach we present three real---life examples in the last section.