Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CHARACTERIZATION OF FINITE COLORED SPACES WITH CERTAIN CONDITIONS

  • Hirasaka, Mitsugu (Department of Mathematics College of Sciences Pusan National University) ;
  • Shinohara, Masashi (Department of Education Faculty of Education Shiga University)
  • Received : 2018.02.06
  • Accepted : 2019.02.27
  • Published : 2019.05.01
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Abstract

A colored space is a pair (X, r) of a set X and a function r whose domain is $\(^X_2\)$. Let (X, r) be a finite colored space and $Y,\;Z{\subseteq}X$. We shall write $Y{\simeq}_rZ$ if there exists a bijection $f:Y{\rightarrow}Z$ such that r(U) = r(f(U)) for each $U{\in}\({^Y_2}\)$ where $f(U)=\{f(u){\mid}u{\in}U\}$. We denote the numbers of equivalence classes with respect to ${\simeq}_r$ contained in $\(^X_i\)$ by $a_i(r)$. In this paper we prove that $a_2(r){\leq}a_3(r)$ when $5{\leq}{\mid}X{\mid}$, and show what happens when equality holds.

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Acknowledgement

Supported by : Pusan National University

References

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