# Algebraic constructions of groupoids for metric spaces

## Main Article Content

## Abstract

Given a groupoid $(X,*)$ and a real-valued function $d: X\to {\bf R}$, a new (derived) function $\Phi(X,*)(d)$ is defined as $[\Phi(X,*)(d)](x, y):= d(x*y) + d(y*x)$ and thus $\Phi(X,*): {\bf R}^X \to {\bf R}^{X^2}$ as well, where ${\bf R}$ is the set of real numbers. The mapping $\Phi(X,*)$ is an {\bf R}-linear transformation also. Properties of groupoids $(X,*)$, functions $d: X\to {\bf R}$, and linear transformations $\Phi(X,*)$ 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., $\Phi(X,*)(d) : X^2 \to X$ is a metric on $X$ for some $d: X\to {\bf R}$.

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## References

[1] P. J. Allen, H. S. Kim and J. Neggers, On companion d-algebras, Math. Slovaca 57 (2) (2007), 93–106. https://dx.doi.org/10.2478/s12175-007-0001-z Google Scholar

[2] A. Iorgulescu, Algebras of Logic as BCK algebras, Editura ASE, Bucharest, 2008. Google Scholar

[3] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa, Korea, 1994. Google Scholar

[4] Gh. Moghaddasi, Sequentially injective and complete acts over a semigroup, J. Nonlinear Sci. Appl. 5 (5) (2012), 345–349. https://dx.doi.org/10.22436/jnsa.005.05.04 Google Scholar

[5] L. Nebesky, Travel groupoids, Czech. Math. J. 56 (2) (2006), 659–675. https://dx.doi.org/10.1007/s10587-006-0046-0 Google Scholar

[6] J. Neggers and H. S. Kim, On d-algebras, Math. Slovaca 49 (1) (1999), 19–26. Google Scholar

[7] J. Neggers, Y. B. Jun and H. S. Kim, On d-ideals in d-algebras, Math. Slovaca 49 (3) (1999), 243–251. Google Scholar

[8] J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik 54 (1–2) (2002), 21–29. Google Scholar

[9] H. K. Park and H. S. Kim, On quadratic B-algebras, Quasigroups Related Syst. 8 (2001), 67–72. Google Scholar

[10] K. P. R. Sastry, Ch. R. Rao, A. C. Sekhar and M. Balaiah, A fixed point theorem in a lattice ordered semigroup cone valued cone metric spaces, J. Nonlinear Sci. Appl. 6 (4) (2013), 285–292. https://dx.doi.org/10.22436/jnsa.006.04.06 Google Scholar

[11] A. Wronski, BCK-algebras do not form a variety, Math. Japon. 28 (1983), 211–213. Google Scholar

[12] H. Yisheng, BCI-algebras, Science Press, Beijing, 2006. Google Scholar