Ideal extensions of lattices (Q735930)

For a lattice \(L\), a map \(\lambda :L\rightarrow L\) is called a \textsl{translation} of \(L\) if \(\lambda (x\wedge y)=\lambda (x)\wedge y\) for all \(x,y\in L\), and an \textsl{inner translation} if \(\lambda (x)=t\wedge x\) for a fixed \(t\in L\). If \(L\) is a lattice, \(K\) is a lattice with \(0\) and \( L\cap K^{\ast }=\emptyset\) (\(K^*=K\setminus\{0\}\)), then a lattice \(V\) is called an \textsl{ideal extension} of \(L\) by \(K\) if there is an ideal \(L^{\prime }\) in \(V\) which is isomorphic to \(L\) and the Rees quotient \(V|L^{\prime }\) is isomorphic to \(K\). In this paper, all the ideal extensions of \(L\) by \(K\) are constructed in such a way that also, conversely, each lattice \(V\) which is an extension of \(L\) by \(K\) can be so constructed. The proof of this main result covers 8 pages. An illustrative example is given at the end. The Rees quotient is defined in Lemma 2.