On connections between information systems, rough sets and algebraic logic

Stephen Comer

Banach Center Publications (1993)

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

Abstract

top
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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.