Turkish Journal of Mathematics
DOI
10.3906/mat-2012-94
Abstract
Let $\mathcal P_{n}$ be the partial transformation semigroup on $X_{n}=\{1,2,\ldots ,n\}$. In this paper, we find the left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal P_{n}$, and their numbers. For $n \geq 3$, we define an undirected graph $\Gamma(\mathcal P_{n})$ associated with $\mathcal P_{n}$ whose vertices are the two sided zero-divisors of $\mathcal P_{n}$ excluding the zero element $\theta$ of $\mathcal P_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta =\beta\alpha$. First, we prove that $\Gamma(\mathcal P_{n})$ is a connected graph, and find the diameter, girth, domination number and the degrees of the all vertices of $\Gamma(\mathcal P_{n})$. Furthermore, we give lower bounds for clique number and chromatic number of $\Gamma(\mathcal P_{n})$.
Keywords
Partial transformation semigroup, zero-divisor graph, clique number, chromatic number
First Page
2323
Last Page
2330
Recommended Citation
TOKER, KEMAL
(2021)
"Zero-divisor graphs of partial transformation semigroups,"
Turkish Journal of Mathematics: Vol. 45:
No.
5, Article 34.
https://doi.org/10.3906/mat-2012-94
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss5/34
