On semigroup presentations and Adian graphs

Let PP be a semigroup presentation of the form〈a1,…,an∣w1=a1,…,wn=an〉.〈a1,…,an∣w1=a1,…,wn=an〉.In this paper we consider the semigroup defined by PP and its Adian graphs. We show that if both Adian graphs of PP are connected and if one of the Adian graphs of PP is a cycle graph then PP defines a group.