The ranks of planarity for~varieties~of~commutative~semigroups
Prikladnaâ diskretnaâ matematika, no. 4 (2016), pp. 50-64
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We study the concept of the planarity rank suggested by L. M. Martynov for semigroup varieties. Let $V$ be a variety of semigroups. If there is a natural number $r\geq 1$ that all $V$-free semigroups of ranks $\leq r$ allow planar Cayley graphs and the $V$-free semigroup of a rank $r+1$ doesn't allow planar Cayley graph, then this number $r$ is called the planarity rank for variety $V$. If such a number $r$ doesn't exist, then we say that the variety $V$ has the infinite planarity rank. We prove that a non-trivial variety of commutative semigroups either has the infinite planarity rank and coincides with the variety of semigroups with the zero multiplication or has a planarity rank $1$, $2$ or $3$. These estimates of planarity ranks for varieties of commutative semigroups are achievable.
Keywords:
semigroup, Cayley graph of semigroup, variety of semigroups, free semigroup of variety, planarity rank for semigroup variety, commutative semigroup, variety of commutative semigroups, planarity rank for variety of commutative semigroups.
@article{PDM_2016_4_a3,
author = {D. V. Solomatin},
title = {The ranks of planarity for~varieties~of~commutative~semigroups},
journal = {Prikladna\^a diskretna\^a matematika},
pages = {50--64},
publisher = {mathdoc},
number = {4},
year = {2016},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/PDM_2016_4_a3/}
}
D. V. Solomatin. The ranks of planarity for~varieties~of~commutative~semigroups. Prikladnaâ diskretnaâ matematika, no. 4 (2016), pp. 50-64. http://geodesic.mathdoc.fr/item/PDM_2016_4_a3/