### Abstract

We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.

Original language | English |
---|---|

Pages (from-to) | 1-14 |

Number of pages | 14 |

Journal | Fundamenta Informaticae |

Volume | 111 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- adjunction
- approximation spaces
- complete lattices
- equivalence of categories
- rough sets

### ASJC Scopus subject areas

- Information Systems
- Computational Theory and Mathematics
- Theoretical Computer Science
- Algebra and Number Theory

### Cite this

*Fundamenta Informaticae*,

*111*(1), 1-14. https://doi.org/10.3233/FI-2011-550

**Double approximation and complete lattices.** / Haruna, Taichi; Gunji, Yukio.

Research output: Contribution to journal › Article

*Fundamenta Informaticae*, vol. 111, no. 1, pp. 1-14. https://doi.org/10.3233/FI-2011-550

}

TY - JOUR

T1 - Double approximation and complete lattices

AU - Haruna, Taichi

AU - Gunji, Yukio

PY - 2011

Y1 - 2011

N2 - We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.

AB - We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.

KW - adjunction

KW - approximation spaces

KW - complete lattices

KW - equivalence of categories

KW - rough sets

UR - http://www.scopus.com/inward/record.url?scp=84856430847&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856430847&partnerID=8YFLogxK

U2 - 10.3233/FI-2011-550

DO - 10.3233/FI-2011-550

M3 - Article

VL - 111

SP - 1

EP - 14

JO - Fundamenta Informaticae

JF - Fundamenta Informaticae

SN - 0169-2968

IS - 1

ER -