Theoretical Backgrounds of Applied Discrete Mathematics
The ranks of planarity for varieties of commutative semigroups
D. V. Solomatin
Omsk State Pedagogical University, Omsk, Russia
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.
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.
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D. V. Solomatin, “The ranks of planarity for varieties of commutative semigroups”, Prikl. Diskr. Mat., 2016, no. 4(34), 50–64
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\paper The ranks of planarity for~varieties~of~commutative~semigroups
\jour Prikl. Diskr. Mat.
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