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 PDF Compressed Postscript Version to read ISSN 0862-9544 (Printed edition) Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295111 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © 2009, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE |