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This article is cited in 1 scientific paper (total in 1 paper)
Theoretical Backgrounds of Applied Discrete Mathematics
The ranks of planarity for varieties of commutative semigroups
D. V. Solomatin Omsk State Pedagogical University, Omsk, Russia
Abstract:
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.
Citation:
D. V. Solomatin, “The ranks of planarity for varieties of commutative semigroups”, Prikl. Diskr. Mat., 2016, no. 4(34), 50–64
Linking options:
https://www.mathnet.ru/eng/pdm567 https://www.mathnet.ru/eng/pdm/y2016/i4/p50
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Abstract page: | 320 | Full-text PDF : | 271 | References: | 397 |
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