Abstract: IMTL and its related logic algebras are investigated in depth again and some new useful results are obtained. Firstly, the IMTL*-algebras are introduced by strengthening the conditions of IMTL-algebra, and the existence of IMTL*-algebras is explained by establishing structures of IMTL*-algebra. Secondly, an algebra structure is established that is IMTL-algebra and not IMTL*-algebra, which explains that IMTL*-algebras are truely strengthened form of IMTL-algebras, and  based on IMTL-algebras some equivalent characterization of IMTL*-algebras are given. At last, the pure implication representation of IMTL*-algebras is obtained. The results of this paper are necessary complementarity to the methods and contents of logic algebras.