Direct products of cyclic semigroups with zero, admitting outerplanar and generalized outerplanar Cayley graphs

D. V. Solomatin

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 90 (2024), pp. 40-49

Résumé

The article presents the characteristic properties of direct products of semigroups with zero admitting outerplanar Cayley graphs, as well as their generalizations in the defining relations of copresentation. Theorem 1. A finite semigroup $S$ with zero that is a direct product of nontrivial cyclic semigroups with zero admits an outerplanar Cayley graph if and only if one of the following conditions holds: 1) $S \cong \langle a\mid a^3 = a^2\rangle^0 \times\langle b \mid b^{h+1}=b^h\rangle^0$ where $h$ is a natural number and $h4$; 2) $S \cong\langle a_0\mid a_0^{r+1}= a_0^r\rangle \times \prod_{i=1}^ n \langle a_i \mid a_i^{2+1}= a_i^2\rangle$ where $r$ and $n$ are natural numbers and $r\leqslant 2$; or $r = 3$, $n = 1$; 3) $S \cong \langle a\mid a^{r+m}=a^r\rangle^{+0}\times \langle b\mid b_2=b\rangle^{+0}$ where $r$ and $m$ are natural numbers and $m \leqslant 2$; 4) $S \cong \langle a_0\mid a_0^{r+1}= a_0^r\rangle \times \prod_{i=1}^n \langle a_i\mid a_i^2= a_i\rangle^{+0}$ where $n = 1$; or $r = 1$, $n = 2$. Theorem 2. A finite semigroup $S$ with zero that is a direct product of nontrivial cyclic semigroups with zero admits a generalized outerplanar Cayley graph if and only if one of the following conditions holds: 1) $S \cong \langle a\mid a^{r+m}=a^r\rangle^0\times \langle b\mid b^{h+t}=b^h\rangle^0$ where for natural numbers $r, m, h, t$ one of the following restrictions is satisfied: 1.1) $r=2$, $m=1$, $h4$, $t=1$; 1.2) $r=3$, $m=1$, $h=3$, $t=1$; 2) $S \cong \langle a_0\mid a_0^{r+1}=a_0^r\rangle\times\prod_{i=1}^n \langle a_i\mid a_i^{2+1}=a_i^2\rangle$ where $r$ and $n$ are natural numbers and $r \leqslant 3$; 3.1) $S \cong\langle a\mid a^{2+1}= a^2\rangle \times \langle b\mid b^{2+1}= b^2\rangle^{+0}$; 3.2) $S \cong\langle a\mid a^{r+m}= a^r\rangle^{+0} \times \langle b \mid b^2= b\rangle^{+0}$ where $r$ and $m$ are natural numbers and $m\leqslant 2$; 4) $S \cong \langle a_0\mid a_0^{r+1}=a_0^r\rangle\times\prod_{i=1}^n \langle a_i\mid a_i^2=a_i\rangle^{+0}$ where $n=1$; or $r=1$, $n=2$.

Détail

Keywords: right Cayley graphs of semigroups, planar graphs, semigroups with zero, direct products of semigroups, outerplanar graphs.