Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ALGEBRAIC CONSTRUCTIONS OF GROUPOIDS FOR METRIC SPACES

  • Se Won Min (Department of Mathematics, Hanyang University) ;
  • Hee Sik Kim (Department of Mathematics, Hanyang University) ;
  • Choonkil Park (Department of Mathematics, Research Institute for Convergence of Basicl Sciences, Hanyang University)
  • Received : 2024.06.05
  • Accepted : 2024.09.12
  • Published : 2024.09.30
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Abstract

Given a groupoid (X, *) and a real-valued function d : X → R, a new (derived) function Φ(X, *)(d) is defined as [Φ(X, *)(d)](x, y) := d(x * y) + d(y * x) and thus Φ(X, *) : RX → RX2 as well, where R is the set of real numbers. The mapping Φ(X, *) is an R-linear transformation also. Properties of groupoids (X, *), functions d : X → R, and linear transformations Φ(X, *) interact in interesting ways as explored in this paper. Because of the great number of such possible interactions the results obtained are of necessity limited. Nevertheless, interesting results are obtained. E.g., if (X, *, 0) is a groupoid such that x * y = 0 = y * x if and only if x = y, which includes the class of all d/BCK-algebras, then (X, *) is *-metrizable, i.e., Φ(X, *)(d) : X2 → X is a metric on X for some d : X → R.

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References

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