ALGEBRAIC CONSTRUCTIONS OF GROUPOIDS FOR METRIC SPACES

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 gro...

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Bibliographic Details
Published in:Korean Journal of mathematics 2024, Vol.32 (3), p.533-544
Main Authors: Se Won Min, Hee Sik Kim, Choonkil Park
Format: Article
Language:Korean
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Summary: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.
ISSN:1976-8605
2288-1433