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In this work, we define (amply) generalized supplemented lattices and investigate some properties of these lattices. In this paper, all lattices are complete modular lattices with the smallest element 0 and the greatest element 1. Let L be a lattice, 1 D a 1 _ a 2 _ : : : _ a n and the quotient sublattices a 1 =0, a 2 =0,.. . , a n =0 be generalized supplemented, then L is generalized supplemented. If L is an amply generalized supplemented lattice, then for every a 2 L, the quotient sublattice 1=a is amply generalized supplemented.

In this work, cofinitely ⊕−supplemented and strongly cofinitely ⊕−supplemented lattices are defined and investigated some properties of these lattices. Let L be a lattice and 1 = ⊕ i∈I a i with a i ∈ L. If a i /0 is cofinitely ⊕−supplemented for every i ∈ I, then L is also cofinitely ⊕−supplemented. Let L be a distributive lattice and 1 = a 1 ⊕ a 2 with a 1 , a 2 ∈ L. If a 1 /0 and a 2 /0 are strongly cofinitely ⊕−supplemented, then L is also strongly cofinitely ⊕−supplemented. Let L be a lattice. If every cofinite element of L lies above a direct summand in L, then L is cofinitely ⊕−supplemented.

In this work, we define Rad-oplus-supplemented and strongly Rad-oplus-supplemented lattices and give some properties of these lattices. We generalize some properties of Rad-oplus-supplemented modules to lattices.

In this work, we define Rad− ⊕ −supplemented and strongly Rad− ⊕ −supplemented lattices and give some properties of these lattices. We generalize some properties of Rad− ⊕ −supplemented modules to lattices. Let L be a lattice and 1 = a 1 ⊕a 2 ⊕. . .⊕a n with a 1 , a 2 , . . . , a n ∈ L. If a i /0 is Rad− ⊕ − supplemented for every i = 1, 2, . . . , n, then L is also Rad− ⊕ − supplemented. Let L be a distributive Rad−⊕−supplemented lattice. Then 1/u is Rad−⊕−supplemented for every u ∈ L. We also define completely Rad− ⊕ −supplemented lattices and prove that every Rad− ⊕ −supplemented lattice with SSP property is completely Rad− ⊕ − supplemented.