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On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

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Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University (Rangsit Campus), Pathum Thani 12121, Thailand
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Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
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Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
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Author to whom correspondence should be addressed.
Mathematics 2018, 6(5), 76; https://doi.org/10.3390/math6050076
Received: 10 April 2018 / Revised: 5 May 2018 / Accepted: 7 May 2018 / Published: 9 May 2018
Let K n be a complete graph on n vertices. Denote by S K n the set of all subgraphs of K n . For each G , H S K n , the ring sum of G and H is a graph whose vertex set is V ( G ) V ( H ) and whose edges are that of either G or H, but not of both. Then S K n is a semigroup under the ring sum. In this paper, we study Green’s relations on S K n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of S K n . Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on S K n and consider minimal elements, maximal elements and covering elements of S K n under this order. View Full-Text
Keywords: complete graph; Green’s relations; ideal; natural order; maximal subsemigroup; maximal congruence complete graph; Green’s relations; ideal; natural order; maximal subsemigroup; maximal congruence
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Chaiya, Y.; Pookpienlert, C.; Nupo, N.; Panma, S. On the Semigroup Whose Elements Are Subgraphs of a Complete Graph. Mathematics 2018, 6, 76.

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