Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a,b} of vertices such that a b and ab=ba. Denote by T(X) the semigroup of full transformations on a finite set X. Let J be any ideal of T(X) such that J is different from the ideal of constant transformations on X.  prove that if |X|>=4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand,  prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n.  also study the left paths in G(S), that is, paths a"1-a"2-...-a"m such that a"1 a"m and a"1a"i=a"ma"i for all i@?{1,...,m}.  prove that for every positive integer n>=2, except n=3, there exists a semigroup whose shortest left path has length n. As a corollary,  use the previous results to solve a purely algebraic old problem posed by B.M. Schein.
['João Araújo', 'Michael K. Kinyon', 'Janusz Konieczny']
I am writing a research paper on a idea Minimal paths in the commuting graphs of semigroups
[{"from": "human", "value": "I am writing a research paper on a idea Minimal paths in the commuting graphs of semigroups"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a,b} of vertices such that a b and ab=ba. Denote by T(X) the semigroup of full transformations on a finite set X. Let J be any ideal of T(X) such that J is different from the ideal of constant transformations on X.  prove that if |X|>=4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand,  prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n.  also study the left paths in G(S), that is, paths a\"1-a\"2-...-a\"m such that a\"1 a\"m and a\"1a\"i=a\"ma\"i for all i@?{1,...,m}.  prove that for every positive integer n>=2, except n=3, there exists a semigroup whose shortest left path has length n. As a corollary,  use the previous results to solve a purely algebraic old problem posed by B.M. Schein."}]