Geodesic
Groupoids assigned to relational systems
Mathematica Bohemica, Tome 138 (2013) no. 1, pp. 15-23
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By a relational system we mean a couple $(A,R)$ where $A$ is a set and $R$ is a binary relation on $A$, i.e.\ $R\subseteq A\times A$. To every directed relational system $\mathcal {A}=(A,R)$ we assign a groupoid ${\mathcal G}({\mathcal A})=(A,\cdot )$ on the same base set where $xy=y$ if and only if $(x,y)\in R$. We characterize basic properties of $R$ by means of identities satisfied by ${\mathcal G}({\mathcal A})$ and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.
By a relational system we mean a couple $(A,R)$ where $A$ is a set and $R$ is a binary relation on $A$, i.e.\ $R\subseteq A\times A$. To every directed relational system $\mathcal {A}=(A,R)$ we assign a groupoid ${\mathcal G}({\mathcal A})=(A,\cdot )$ on the same base set where $xy=y$ if and only if $(x,y)\in R$. We characterize basic properties of $R$ by means of identities satisfied by ${\mathcal G}({\mathcal A})$ and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.
DOI : 10.21136/MB.2013.143226
Classification : 08A02, 20N02
Keywords: relational system; groupoid; directed system; $g$-homomorphism 
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Chajda, Ivan; Länger, Helmut. Groupoids assigned to relational systems. Mathematica Bohemica, Tome 138 (2013) no. 1, pp. 15-23. doi: 10.21136/MB.2013.143226

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