Algebraic constructions of groupoids for metric spaces
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초록
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 $d: X\to {\bf R}$.
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 $d: X\to {\bf R}$.
참고문헌 (12)
- P. J. Allen, "On companion d-algebras", Math. Slovaca, 57(2), pp.93~106 (2007)[DOI]
- A. Iorgulescu, "Algebras of Logic as BCK algebras" (2008), Editura ASE
- J. Meng, "BCK-algebras" (1994), Kyungmoon Sa
- Gh. Moghaddasi, "Sequentially injective and complete acts over a semigroup", J. Nonlinear Sci. Appl, 5(5), pp.345~349 (2012)[DOI]
- L. Nebesky, "Travel groupoids", Czech. Math. J, 56(2), pp.659~675 (2006)[DOI]
- J. Neggers, "On d-algebras", Math. Slovaca, 49(1), pp.19~26 (1999)
- J. Neggers, "On d-ideals in d-algebras", Math. Slovaca, 49(3), pp.243~251 (1999)
- J. Neggers, "On B-algebras", Mate. Vesnik, 54(1-2), pp.21~29 (2002)
- H. K. Park, "On quadratic B-algebras", Quasigroups Related Syst, 8, pp.67~72 (2001)
- K. P. R. Sastry, "A xed point theorem in a lattice ordered semigroup cone valued cone metric spaces", J. Nonlinear Sci. Appl, 6(4), pp.285~292 (2013)[DOI]
- A. Wronski, "BCK-algebras do not form a variety", Math. Japon, 28, pp.211~213 (1983)
- H. Yisheng, "BCI-algebras" (2006), Science Press