Noetherian Lattices

The notation and terminology used here are introduced in the following papers: [18], [13], [17], [14], [19], [7], [1], [8], [6], [20], [3], [9], [2], [10], [15], [16], [5], [11], [4], and [12]. Let us observe that there exists a lattice which is finite. Let us mention that every lattice which is finite is also complete. Let L be a lattice and let D be a subset of the carrier of L. The functor D yields a subset of Poset(L) and is defined by: (Def. 1) D = {d; d ranges over elements of the carrier of L: d ∈ D}. Let L be a lattice and let D be a subset of the carrier of Poset(L). The functor D yielding a subset of the carrier of L is defined by: (Def. 2) D = {d; d ranges over elements of Poset(L): d ∈ D}. Let L be a finite lattice. Note that Poset(L) is well founded. Let L be a lattice. We say that L is noetherian if and only if: (Def. 3) Poset(L) is well founded. We say that L is co-noetherian if and only if: (Def. 4) Poset(L)` is well founded. One can verify the following observations: ∗ there exists a lattice which is noetherian and upper-bounded, ∗ there exists a lattice which is noetherian and lower-bounded, and ∗ there exists a lattice which is noetherian and complete.