Algrbraic properties and structures of intuitionistic L−fuzzy multiset finite automata (ILFMA) are discussed through congruences on a semigroup in this paper. Firstly, we put forward the notion of the intuitionistic L−fuzzy compatible relation, the compatible monoid associated to the intuitionistic L−fuzzy compatible relation can be effectively constructed, and we construct two finite monoids through two different congruence relations on a given ILFMA, then we also prove that they are isomorphic. Furthermore, using the quotient structure of ILFMA, algebraic properties of intuitionistic L−fuzzy multiset transformation semigroup are discussed. According to intuitionistic L−admissible relation and homomorphism of ILFMA, we show that there is a bijective correspondence between an ILFMA and the quotient structure of another ILFMA.