THE GRAPH OF DIAGRAM GROUPS FROM DIRECT PRODUCT OF TWO FREE SEMIGROUPS

In this paper, we will discuss the diagram groups from direct product of two  free semigroups $\left[X,Y|xy=yx (x \in X, y \in Y)\right]$  and their graphs will be presented. The graphs  obtained  are related to the word length and  the exponent sum. The exponent sum of  $X$ is three and the exponent sum of $Y$ is  $n ( N)$.  Finally, the number of  generator of the diagram group can be determined.