Algebraic constructions of groupoids for metric spaces

Abstract

Given a groupoid $(X,\ast )$ and a real-valued function $d:X\to \mathbf{R}$, a new (derived) function $\mathrm{\Phi }(X,\ast )(d)$ is defined as $[\mathrm{\Phi }(X,\ast )(d)](x,y):=d(x\ast y)+d(y\ast x)$ and thus $\mathrm{\Phi }(X,\ast ):{\mathbf{R}}^X\to {\mathbf{R}}^{X^2}$ as well, where $\mathbf{R}$ is the set of real numbers. The mapping $\mathrm{\Phi }(X,\ast )$ is an {\bf R}-linear transformation also. Properties of groupoids $(X,\ast )$, functions $d:X\to \mathbf{R}$, and linear transformations $\mathrm{\Phi }(X,\ast )$ 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 $(X,\ast ,0)$ is a groupoid such that $x\ast y=0=y\ast x$ if and only if $x=y$, which includes the class of all $d/BCK$-algebras, then $(X,\ast )$ is $\ast$-metrizable, i.e., $\mathrm{\Phi }(X,\ast )(d):X^2\to X$ is a metric on $X$ for some $d:X\to \mathbf{R}$.