Cilt 22, Sayı 22, Sayfalar 62 - 77 2017-07-11
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## The $x$-divisor pseudographs of a commutative groupoid

#### John D. LaGrange [1]

##### 98 35

The notion of a zero-divisor graph is considered for commutative groupoids with zero. Moufang groupoids and certain medial groupoids with zero are shown to have connected zero-divisor graphs of diameters at most four and three, respectively. As $x$ ranges over the elements of a commutative groupoid $\mB$ (not necessarily with zero), a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of $\mB$ and vertices $a$ and $b$ are adjacent if and only if $ab=x$. These systems are completely characterized as being partitions of complete pseudographs $\overline{K}_{n}$ whose parts are indexed by the vertices of $\overline{K}_{n}$. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it (categorically) isomorphic to the category of commutative groupoids, thereby combinatorializing the theory of commutative groupoids. Also, concepts of congruence" and direct product" that are compatible with those in the category of commutative groupoids are established for these systems of pseudographs.

Groupoid,zero-divisor graph
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Konular Matematik ve İstatistik Makaleler Yazar: John D. LaGrangeE-posta: lagrangej@lindsey.edu
 Bibtex @araştırma makalesi { ieja325926, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, address = {Hacettepe Üniversitesi}, year = {2017}, volume = {22}, pages = {62 - 77}, doi = {10.24330/ieja.325926}, title = {The \$x\$-divisor pseudographs of a commutative groupoid}, language = {en}, key = {cite}, author = {LaGrange, John D.} } APA LaGrange, J . (2017). The $x$-divisor pseudographs of a commutative groupoid. International Electronic Journal of Algebra, 22 (22), 62-77. DOI: 10.24330/ieja.325926 MLA LaGrange, J . "The $x$-divisor pseudographs of a commutative groupoid". International Electronic Journal of Algebra 22 (2017): 62-77 Chicago LaGrange, J . "The $x$-divisor pseudographs of a commutative groupoid". International Electronic Journal of Algebra 22 (2017): 62-77 RIS TY - JOUR T1 - The $x$-divisor pseudographs of a commutative groupoid AU - John D. LaGrange Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.325926 DO - 10.24330/ieja.325926 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 62 EP - 77 VL - 22 IS - 22 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.325926 UR - http://dx.doi.org/10.24330/ieja.325926 Y2 - 2018 ER - EndNote %0 International Electronic Journal of Algebra The $x$-divisor pseudographs of a commutative groupoid %A John D. LaGrange %T The $x$-divisor pseudographs of a commutative groupoid %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 22 %N 22 %R doi: 10.24330/ieja.325926 %U 10.24330/ieja.325926