| | | |

The $x$-divisor pseudographs of a commutative groupoid

John D. LaGrange [1]

145 94

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
• D. F. Anderson, M. C. Axtell and J. A. Stickles, Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson, Eds.), Springer-Verlag, New York, (2011), 23-45.
• D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
• D. F. Anderson and S. B. Mulay, On the girth and diameter of a zero-divisor graph, J. Pure Appl. Algebra, 210(2) (2007), 543-550.
• M. Axtell, J. Coykendall and J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra, 33(6) (2005), 2043- 2050.
• I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
• R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, Berlin, 1958.
• J. H. Conway, A simple construction for the Fischer-Griess monster group, Invent. Math., 79(3) (1985), 513-540.
• J. Coykendall, S. Sather-Wagstaff, L. Sheppardson and S. Spiroff, On zero divisor graphs, in Progress in Commutative Algebra II: Closures, Finiteness and Factorization (C. Francisco et al., Eds.),Walter de Gruyter, Berlin, (2012), 241-299.
• F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65(2) (2002), 206-214.
• R. Halas and M. Jukl, On Beck's coloring of posets, Discrete Math., 309(13) (2009), 4584-4589.
• B. A. Hausmann and O. Ore, Theory of quasi-groups, Amer. J. Math., 59(4) (1937), 983-1004.
• T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
• J. Jezek and T. Kepka, A note on medial division groupoids, Proc. Amer. Math. Soc., 119(2) (1993), 423-426.
• V. Joshi and A. Khiste, On the zero divisor graphs of pm-lattices, Discrete Math., 312 (2012), 2076-2082.
• A. D. Keedwell, Uniform P-circuit designs, quasigroups, and Room squares, Utilitas Math., 14 (1978), 141-159.
• U. Knauer, Algebraic Graph Theory: Morphisms, Monoids and Matrices, De Gruyter Studies in Mathematics, 41, Walter de Gruyter & Co., Berlin, 2011.
• A. Kotzig, Groupoids and partitions of complete graphs, in Combinatorial Structures and their Applications (Proc. Colloq. Calgary 1969), Gordon and Breach, New York, (1970), 215-221.
• D. Lu and T. Wu, The zero-divisor graphs of posets and an application to semigroups, Graphs Combin., 26(6) (2010), 793-804.
• T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301(1) (2006), 174-193.
• B. V. Novikov, On decomposition of commutative Moufang groupoids, Quasi- groups Related Systems, 16(1) (2008), 97-101.
• S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Com- mut. Rings, 1(4) (2002), 203-211.
• I. M. Wanless and E. C. Ihrig, Symmetries that latin squares inherit from 1-factorizations, J. Combin. Des., 13(3) (2005), 157-172.
Konular Matematik Makaleler Yazar: John D. LaGrange
 Bibtex @araştırma makalesi { ieja325926, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2017}, volume = {22}, pages = {62 - 77}, doi = {10.24330/ieja.325926}, title = {The \$x\$-divisor pseudographs of a commutative groupoid}, 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 ISNAD LaGrange, John D. . "The $x$-divisor pseudographs of a commutative groupoid". International Electronic Journal of Algebra 22 / 22 (Temmuz 2017): 62-77. http://dx.doi.org/10.24330/ieja.325926