### On large arbitrary left path within a semigroup

#### Abstract

This paper is about the construction of semigroups $S$ from some given graph $G$. Let $S$ be a finite non-commutative semigroup, its commuting graph, denoted by $G(S)$, is a simple graph (which has no loops and multiple edges) whose sets of vertices are elements of $S$ and whose sets of edges are those elements of $S$ which commute with other elements i.e. for any $a,b \in S$ such that $ab=ba$ for $a \neq b$.For some non empty finite set $X$,denote $T(X)$ by semigroup of full transformations and $I_{r}$ by ideal of $T(X)$ whose rank is less than or equals to $r$. Let $a_{1}-a_{2}-a_{3}-\ldots-a_{m}$ be a path in $G(S)$, this path is said to be left path or $l-$ path if

$$a_{1}a_{i}=a_{m}a_{i} for i\in\{1,2,3,\ldots m\}$$

In this paper, we construct semigroup $S$ of a complete bipartite graph $K _{n,m}$ and find maximum length of $l-$ path in its commuting graph $G(S)$. Moreover,we see that such type of semigroups have knit degree $2$.**Published:**2017-03-21

**How to Cite this Article:**Ajmal Ali, Zahid Raza, On large arbitrary left path within a semigroup, Journal of Semigroup Theory and Applications, Vol 2017 (2017), Article ID 3 Copyright © 2017 Ajmal Ali, Zahid Raza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Journal of Semigroup Theory and Applications

ISSN 2051-2937

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