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Algebraic Structures on Finite Sets

  • Jack E. Graver
  • Mark E. Watkins
Part of the Graduate Texts in Mathematics book series (GTM, volume 54)

Abstract

In IB we introduced the characteristic functions c S for subsets S of a set U and proved (IB2) that the function c S S is a bijection between K U and P(U). Subsequently it was to be verified (Exercise IB3) that this same bijection made the assignments c S + c T S + T and c S c T ST. We have thereby that (P(U), +, ∩) is “algebra-isomorphic” to the commutative algebra (K U , +, ·), and hence (P(U), +, ∩) is a commutative algebra over the field K. In particular, (P(U), +) is a vector space over K, while (P(U), +, ∩) is a commutative ring; ∅ is the additive identity and U itself is the multiplicative identity. For the present we shall be concerned only with the vector space structure.

Keywords

Partial Order Bipartite Graph Distributive Lattice Algebraic Structure Direct Summand 
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.

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Copyright information

© Springer-Verlag, New York Inc. 1977

Authors and Affiliations

  • Jack E. Graver
    • 1
  • Mark E. Watkins
    • 1
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA

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