Some properties of r-small elements in lattices
Narmin Bayat, Celıl Nebıyev · Miskolc mathematical notes/Mathematical notes · 2026
In this paper, all lattices are complete modular lattices with the greatest element 1 and the smallest element 0 . Let L be a lattice, a ∈ L and a ≤ r ( L ) . If a ≪ r ( L ) ∕ 0 , then a is called an r-small ( or r-superfluous ) element of L and denoted by a ≪ r L . In this work, some properties of these elements are investigated. This concept is a generalization of an r-small submodule of any module. It is clear that every r-small element is small. But the converse of this statement is not true in general. Let L be a lattice, a ∈ r ( L ) ∕ 0 and r ( L ) be a supplement element in L . Then a ≪ L if and only if a ≪ r L . Let L be a lattice, a ∈ L and b ≪ r L . Then a ≪ r L if and only if a ∨ b ≪ r 1 ∕ b