On the Lattice of all Convex Lattice ordered Subnear-rings of Lattice-ordered Near-rings
Kai Xie · Journal of Hunan Educational Institute · 2000
In this paper, the basic property and the lattice of all convex lattice ordered near ordered subnear rings of lattice ordered near rings are discussed, and the following theorems are given and proved:A) Let L be a lattice ordered near ring, the following are equivalent: 1) M is a convex lattice ordered subnear ring; 2) M is a convex and directive; 3) The right coset of M R(M) is a distributive lattice, moreover m 1,m 2∈L,(M+m 1)∨(M+m 2)=M+m 1∨m 2,(M+m 1)∧(M+m 2)=M+m 1∧m 2. B) Let L be a lattice ordered near ring, then C(L)=｛M｜M is a convex lattice ordered subnear ring｝ is a Brouwerian lattice, which is a sublattice of the lattice of all subnear rings of L.