There are several equivalent definitions of the class of semigroups which we call rectangular semigroups. We will use the term rectangular semigroup to denote direct products of singular semigroups. A semigroup is called singular if it is a left zero semigroup or a right zero semigroup. A characteristic property of ordinal sums of rectangular semigroups with planar Cayley graphs is known. This article presents the characteristic properties of ordinal sums of rectangular semigroups with outerplanar Cayley graphs. In addition, the possibilities of their generalization to generalized outerplanar Cayley graphs of semigroups in the same class are analyzed. Namely, a necessary and sufficient condition for the existence of an outerplane embedding in the plane or generalized outerplane embedding in the plane of the Cayley graphs of the ordinal sums of rectangular semigroups is proved. The case when the Cayley graphs of such semigroups turn out to be generalized outerplanar, but not outerplanar, is considered in detail. The paper considers generalized outerplanar graphs characterized by Jiří Sedláček.
Keywords: outerplanar graph, semigroups with outerplanar Cayley graphs, generalized outerplanar graphs, Sedláček graphs, semigroups with planar Cayley graphs
Received March 25, 2024
Revised May 4, 2024
Accepted May 13, 2024
Denis Vladimirovich Solomatin, Cand. Sci. (Phys.-Math.), Omsk State Pedagogical University, Omsk, 644099 Russia, e-mail: solomatin_dv@omgpu.ru
REFERENCES
1. Klement E.P., Mesiar R., Pap E. Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford. Semigroup Forum, 2002, vol. 65, pp. 71–82. doi: 10.1007/s002330010127
2. Su Y., Zong W., Mesiarová-Zemánková A. Constructing uninorms via ordinal sums in the sense of A. H. Clifford. Semigroup Forum, 2022, vol. 105, pp. 328–344. doi: 10.1007/s00233-022-10287-1
3. Maschke H. The representation of finite groups, especially of the rotation groups of the regular bodies of three- and four-dimensional space, by Cayley’s color diagrams. Amer. J. Math., 1896, vol. 18, no. 2, pp. 156–194.
4. Knauer K., Knauer U. On planar right groups. Semigroup Forum, 2015, vol. 92, no. 1, pp. 142–157. doi: 10.1007/s00233-015-9688-2
5. Knauer K., Knauer U. Algebraic graph theory: morphisms, monoids and matrices. Ser. De Gruyter studies in mathematics, vol. 41, 2nd rev. and ext. ed., Berlin, Boston, De Gruyter Publ., 2019, 349 p. doi: 10.1515/9783110617368
6. Zhang X. Clifford semigroups with genus zero. In: Proc. Int. Conf. Semigroups, acts and categories with applications to graphs, University of Tartu, 2007, Estonian Math. Soc. Tartu., 2008, vol. 3, pp. 151–160. ISBN: 978–9985–9644–2–2 .
7. Zhu Y. Generalized Cayley graphs of semigroups I. Semigroup Forum, 2012, vol. 84, pp. 131–143. doi: 10.1007/s00233-011-9368-9
8. Solomatin D.V. Researches of semigroups with planar Cayley graphs: results and problems. Prikl. Diskr. Mat., 2021, no. 54, pp. 5–57 (in Russian).
9. Solomatin D.V. Direct products of cyclic monoids admitting outerplanar Cayley graphs and their generalizations. Vestnik TvGU. Ser. Prikladnaya Matematika, 2023, vol. 4, pp. 43–56 (in Russian). doi: 10.26456/vtpmk697
10. Adyan S.I. Algorithmic unsolvability of the recognition problem for certain properties of groups. Dokl. AN SSSR, 1955, vol. 103, no. 4, pp. 533–535 (in Russian).
11. Shevrin L.N. Semigroups. General algebra. Ed. L. A. Skornyakov. Moscow, Nauka Publ., 1991, vol. 2, ch. IV, pp. 11–191 (in Russian). ISBN: 5-02-014427-4 .
12. Harary F. Graph Theory. Advanced book program ser., Boulder, Westview Press, 1994, 288 p.
13. Sedláček J. On a generalization of outerplanar graphs. Časopis Pěst. Mat., 1988, vol. 2, no. 113, pp. 213–218.
14. Ding G., Dziobiak S. Excluded-minor characterization of apex-outerplanar graphs. Graphs and combinatorics, 2016, vol. 32, no. 2, pp. 583–627. doi: 10.1007/s00373-015-1611-9
Cite this article as: D.V. Solomatin. Ordinal sums of rectangular semigroups with outerplanar Cayley graphs and their generalizations. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 251–264.
