caa a22 4500 445345047 CHVBK 20180317142826.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00233-010-9289-z doi (NATIONALLICENCE)springer-10.1007/s00233-010-9289-z Two cancellative commutative congruences andgroupdiagrams [Elektronische Daten] [A. Clifford, P. Cummings, J. Teymouri] Remmers (Adv. Math. 36:283-296, 1980) uses group diagrams in the Euclidean plane to demonstrate how equality in a semigroup S "mirrors” that inside the group G sharing the same presentation withS, when S satisfies Adyan's condition—no cycles in the left/right graphs of the semigroup's presentation. Goldstein and Teymouri (Semigroup Forum 47:299-304, 1993) introduce a conjugacy equivalence relation for semigroups S. By closely examining the geometry of annular group diagrams in the plane, they show how their equivalence relation mirrors conjugacy inside G, for S satisfying Adyan's. In this article we introduce two cancellative commutative congruences. Following their leads, we examine the geometry of group diagrams on closed surfaces of higher genera to demonstrate how these congruences mirror equality inside two naturally associated Abelian quotient groups G/[G,G] and G/G 2, respectively. In these instances we can drop Adyan's condition. Springer Science+Business Media, LLC, 2010 Commutative congruences nationallicence Group diagrams nationallicence Clifford A. Department of Mathematics and Statistics, The College of New Jersey, P.O. Box 7718, 08628, Ewing, NJ, USA aut Cummings P. Educational Opportunities Program, University at Albany, State University of New York, 1400 Washington Avenue, 12222, Albany, NY, USA aut Teymouri J. Department of Mathematics, The College of Saint Rose, 432 Western Avenue, 12203, Albany, NY, USA aut Semigroup Forum Springer-Verlag 82/2(2011-04-01), 338-353 0037-1912 82:2<338 2011 82 233 https://doi.org/10.1007/s00233-010-9289-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00233-010-9289-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Clifford A. Department of Mathematics and Statistics, The College of New Jersey, P.O. Box 7718, 08628, Ewing, NJ, USA aut NATIONALLICENCE 700 1- Cummings P. Educational Opportunities Program, University at Albany, State University of New York, 1400 Washington Avenue, 12222, Albany, NY, USA aut NATIONALLICENCE 700 1- Teymouri J. Department of Mathematics, The College of Saint Rose, 432 Western Avenue, 12203, Albany, NY, USA aut NATIONALLICENCE 773 0- Semigroup Forum Springer-Verlag 82/2(2011-04-01), 338-353 0037-1912 82:2<338 2011 82 233 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer