Combinations of lattice-valued coarse structures and groups

Based on a complete residuated lattice L, we combine the lattice-valued coarse structures and group operations to propose the concept of L-fuzzifying coarse groups. Then we introduce the notion of L-fuzzifying group ideals and establish its one-to-one correspondence with L-fuzzifying coarse groups. Specifically, we examine how L-fuzzifying coarse structures align with the algebraic structures of the supporting group. Finally, we use L-fuzzifying group ideals to characterize a fuzzy coarse equivalence between L-fuzzifying coarse groups, presenting some results derived from the kernel of the group homomorphism.