Periodica Mathematica Hungarica

, Volume 44, Issue 1, pp 111–126

# Sublattices of regular elements

• D. D. Anderson
• E. W. Johnson
• Richard L. Spellerberg II
Article

DOI: 10.1023/A:1014932204184

Anderson, D.D., Johnson, E.W. & Spellerberg II, R.L. Periodica Mathematica Hungarica (2002) 44: 111. doi:10.1023/A:1014932204184

## Abstract

Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$L_f = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\left. {(0:A) = 0} \right\},L_{sr} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|$$ there is a compact element $$X \leqslant A$$ with $$\left. {(0:X) = 0} \right\},L_r = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|$$ there is a principal element $$X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|A = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha$$ is a principal element with (0:X_{\alpha })=0\} . The first three subsets L_{f}, L_{sr}, and L_{r} are augmented filters $$\mathcal{L}^0$$ on L, i.e., $$\mathcal{L}^0 = \mathcal{L} \cup \left\{ 0 \right\}{\text{ where }}\mathcal{L}$$ is a multiplicatively closed subset of $L$ with $A\in $$\mathcal{L}$$ and$B\geq A$with$B\in L$implies$B\in $$\mathcal{L}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module.