p. 47 - 54 Some simple extensions of Eulerian lattices
A. Vethamanickam and R. Subbarayan Received: October 30, 2008;
Revised: August 4, 2009;
Accepted: October 16, 2009
Abstract.
Let L be a lattice. If K is a sublattice of L, then L is called an extension 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
various 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 lattice. 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.
Keywords:
lattices; simple lattices; Eulerian lattices; lattice extension; congruence-preserving extension.
AMS Subject classification:
Primary: 06A06, 06A07, 06B10
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