Ideal extensions of lattices

Following the well known Schreier's extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford–Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice $L$ and a lattice $K$ having a least element, and construct (all) the lattices $V$ which have an ideal $L'$ which is isomorphic to $L$ and the Rees quotient $V|L'$ is isomorphic to $K$. Conversely, we prove that each lattice which is an extension of $L$ by $K$ can be so constructed. An illustrative example is given at the end.