Advertisement

Rough Sets pp 251-272 | Cite as

Algebraic Structures

  • Lech Polkowski
Part of the Advances in Soft Computing book series (AINSC, volume 15)

Abstract

We will be interested here in algebraic structures defined within the realm of ordered sets. These structures play an eminent role in semantics of various logical calculi. In developing a theory of these structures, two ways are possible, either to begin with the most perfect structure i.e. Boolean algebras and relax gradually its requirements descending to less organized structures or to begin with the least perfect structures and gradually add requirements to ascend to more organized structures. Either approach has its merits, and here we settle with the latter so we begin with the minimal sound structure and subsequently add more features.

Keywords

Topological Space Boolean Algebra Distributive Lattice Algebraic Structure Unit Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Works quoted

  1. [Birkhoff67]
    G. Birkhoff, Lattice Theory, AMS, Providence, 1940 (3rd ed., 1967 ).Google Scholar
  2. [Boole847]
    G. Boole, The Mathematical Analysis of Logic, Cambridge, 1847.Google Scholar
  3. [McKinsey—Tarski44]
    J. C. C. McKinsey and A. Tarski, The algebra of topology, Annals of Mathematics, 45 (1944), pp. 141–191.MathSciNetMATHCrossRefGoogle Scholar
  4. [Rasiowa74]
    H. Rasiowa, An Algebraic Approach to Non—Classical Logics, North Holland, 1974.Google Scholar
  5. [Rasiowa—Sikorski63]
    H. Rasiowa and R. Sikorski, The Mathematics of Meta-mathematics, PWN-Polish Scientific Publishers, Warszawa, 1963.Google Scholar
  6. [Schröder895]
    E. Schröder, Algebra der Logic, Leipzig, 1890–1895.Google Scholar
  7. [Stone36]
    M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc., 40 (1936), pp. 37–111.MathSciNetGoogle Scholar
  8. [Tarski38]
    A. Tarski, Der Aussagenkalkill und die Topologie, Fund. Math., 31 (1938), pp. 103–134.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lech Polkowski
    • 1
    • 2
  1. 1.Polish-Japanese Institute of Information TechnologyWarsawPoland
  2. 2.Department of Mathematics and Computer ScienceUniversity of Wormia and MazuryOlsztynPoland

Personalised recommendations