# On connections between information systems, rough sets and algebraic logic

Banach Center Publications (1993)

- Volume: 28, Issue: 1, page 117-124
- ISSN: 0137-6934

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topComer, Stephen. "On connections between information systems, rough sets and algebraic logic." Banach Center Publications 28.1 (1993): 117-124. <http://eudml.org/doc/262784>.

@article{Comer1993,

abstract = {In this note we remark upon some relationships between the ideas of an approximation space and rough sets due to Pawlak ([9] and [10]) and algebras related to the study of algebraic logic - namely, cylindric algebras, relation algebras, and Stone algebras. The paper consists of three separate observations. The first deals with the family of approximation spaces induced by the indiscernability relation for different sets of attributes of an information system. In [3] the family of closure operators defining these approximation spaces is abstractly characterized as a certain type of Boolean algebra with operators. An alternate formulation in terms of a general class of diagonal-free cylindric algebras is given in 1.6. The second observation concerns the lattice theoretic approach to the study of rough sets suggested by Iwiński [6] and the result by J. Pomykała and J. A. Pomykała [11] that the collection of rough sets of an approximation space forms a Stone algebra. Namely, in 2.4 it is shown that every regular double Stone algebra is embeddable into the algebra of all rough subsets of an approximation space. Finally, a notion of rough relation algebra is formulated in Section 3 and a few connections with the study of ordinary relation algebras are established.},

author = {Comer, Stephen},

journal = {Banach Center Publications},

keywords = {approximation space; rough sets; cylindric algebras; relation algebras; Stone algebras},

language = {eng},

number = {1},

pages = {117-124},

title = {On connections between information systems, rough sets and algebraic logic},

url = {http://eudml.org/doc/262784},

volume = {28},

year = {1993},

}

TY - JOUR

AU - Comer, Stephen

TI - On connections between information systems, rough sets and algebraic logic

JO - Banach Center Publications

PY - 1993

VL - 28

IS - 1

SP - 117

EP - 124

AB - In this note we remark upon some relationships between the ideas of an approximation space and rough sets due to Pawlak ([9] and [10]) and algebras related to the study of algebraic logic - namely, cylindric algebras, relation algebras, and Stone algebras. The paper consists of three separate observations. The first deals with the family of approximation spaces induced by the indiscernability relation for different sets of attributes of an information system. In [3] the family of closure operators defining these approximation spaces is abstractly characterized as a certain type of Boolean algebra with operators. An alternate formulation in terms of a general class of diagonal-free cylindric algebras is given in 1.6. The second observation concerns the lattice theoretic approach to the study of rough sets suggested by Iwiński [6] and the result by J. Pomykała and J. A. Pomykała [11] that the collection of rough sets of an approximation space forms a Stone algebra. Namely, in 2.4 it is shown that every regular double Stone algebra is embeddable into the algebra of all rough subsets of an approximation space. Finally, a notion of rough relation algebra is formulated in Section 3 and a few connections with the study of ordinary relation algebras are established.

LA - eng

KW - approximation space; rough sets; cylindric algebras; relation algebras; Stone algebras

UR - http://eudml.org/doc/262784

ER -

## References

top- [1] R. Beazer, The determination congruence on double p-algebras, Algebra Universalis 6 (1976), 121-129. Zbl0353.06002
- [2] S. D. Comer, Representations by algebras of sections over Boolean spaces, Pacific J. Math. 38 (1971), 29-38. Zbl0219.08002
- [3] S. D. Comer, An algebraic approach to the approximation of information, Fund. Inform. 14 (1991), 492-502. Zbl0727.68114
- [4] G. Grätzer, Lattice Theory. First Concepts and Distributive Lattices, W. H. Freeman, San Francisco 1971.
- [5] L. Henkin, J. D. Monk and A. Tarski, Cylindric Algebras, Part I, II, North-Holland, Amsterdam 1985.
- [6] T. B. Iwiński, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35 (1987), 673-683. Zbl0639.68125
- [7] T. Katriňák, Injective double Stone algebras, Algebra Universalis 4 (1974), 259-267. Zbl0302.06022
- [8] I. Németi, Algebraizations of quantifier logics, an introductory overview, preprint, Math. Inst. Hungar. Acad. Sci., 1991. Zbl0772.03033
- [9] Z. Pawlak, Information system - theoretical foundations, Inform. Systems 6 (1981), 205-218. Zbl0462.68078
- [10] Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci. 11 (5) (1982), 341-356. Zbl0501.68053
- [11] J. Pomykała and J. A. Pomykała, The Stone algebra of rough sets, Bull. Polish Acad. Sci. Math. 36 (1988), 495-508. Zbl0786.04008
- [12] H. Werner, Discriminator Algebras, Akademie-Verlag, Berlin 1978.

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