Vol. LXXIX, 1 (2010)
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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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

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Comenius University
842 48 Bratislava, Slovak Republic  

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