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In this paper, we continue our study on the result of Kilibarda, “the diagram group ) , ( w S D is isomorphic to the fundamental group ) ), ( ( 1 w S K π of the ) ( complex 2 S K − with the basepoint w , where S is a monoid presentation and w is positive word over alphabet of S ” to construct of the first ) ( complex N K of diagram group from the semigroup presentation of natural numbers with repeating generators. We will show the diagram group with positive word w 2 ≥ is free of rank 1 2 ) 2 ) 1 ( ( ) , ( + ⎥⎦ ⎤ ⎢⎣ ⎡ − − ⋅ = n m n m n m μ , where n the numbers of generators in ) , ( w N D , 1 , 2 ≥ ≥ m n and ∈ m n, Z + .

The computation of the rank of diagram groups constructed from semigroup presentation