SOME SIMPLE EXTENSIONS OF EULERIAN LATTICES
A. Vethamanickam, R. Subbarayan · 2010
Abstract. Let L be a lattice. If K is a sublattice of L, then L is called an ex-tension of K. Lattice extension concept was elaborately studied by G. Grätzer and E. T. Schmidt in their papers [6], [7], [9], [10]. A lattice L is said to be simple if it has no non-trivial congruences. A finite graded poset P is said to be Eulerian if its Möbius function assumes the value µ(x, y) = (−1)l(x,y) for all x ≤ y in P, where l(x, y) = ρ(y)−ρ(x) and ρ is the rank function on P. In this paper, we exhibit var-ious possible Eulerian extensions which are simple for any given Eulerian lattice L and we prove that there exists a congruence-preserving extension of an Eulerian lat-tice. The cubic extension of a lattice was defined by G. Grätzer and E. T. Schmidt in [11]. We show that the cubic extension becomes a congruence-preserving extension when the lattice is Eulerian.