Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T13:44:10.504Z Has data issue: false hasContentIssue false
  • English
  • Français

Pasting infinite lattices

Part of: Lattices

Published online by Cambridge University Press:  09 April 2009

E. Fried  and
G. Grātzer
E. Fried
Affiliation:
Department of Mathematics, University of ManitobaWinnipeg, Manitoba, R3T 2N2, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In an earlier paper, we investigated for finite lattices a concept introduced by A. Slavik: Let A, B, and S be sublattices of the lattice L, A∩B = S, A∪B = L. Then L pastes A and B together over S, if every amalgamation of A and B over S contains L as a sublattice. In this paper we extend this investigation to infinite lattices. We give several characterizations of pasting; one of them directly generalizes to the infinite case the characterization theorem of A. Day and J. Ježk. Our main result is that the variety of all modular lattices and the variety of all distributive lattices are closed under pasting.

MSC classification

Secondary: 06B05: Structure theory 06B20: Varieties of lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 1 - 21
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Day, A. and Ježek, J., ‘The amalgamation property for varieties of lattices’, Mathematics Report# 1–83, Lakehead University.Google Scholar
[2]Dilworth, R. P. and Hall, M., ‘The embedding problem for modular lattices’, Ann. of Math. (2) 45 (1944), 450–456.Google Scholar
[3]Fried, E. and Grätzer, G., ‘ Notes on tolerance relations of lattices: On a conjecture of R. McKenzie ’, manuscript; to appear in Acta Sci. Math. (Szeged).Google Scholar
[4]Fried, E. and Grätzer, G., ‘ Pasting and modular lattices ’, manuscript (Abstract: Notices Amer. Math. Soc. 87T–06–209); to appear in Proc. Amer. Math. Soc.Google Scholar
[5]Fried, E., Grätzer, G., and Lakser, H., ‘ Projective geometries as cover preserving sublattices’, manuscript (Abstract: Notices Amer. Math. Soc. 88T–06–47); to appear in Algebra Universalis.Google Scholar
[6]Grätzer, G., General lattice theory (Pure and Applied Mathematics Series, Academic Press, New York, 1978; Birkhäuser Verlag, Mathematische Reihe, Band 52, Basel, 1978).Google Scholar
[7]Grätzer, G., Jónsson, B., and Lakser, H., ‘The Amalgamation Property in equational classes of modular lattices’, Pacific J. Math. 45 (1973), 507–524.Google Scholar
[8]Grätzer, G. and Kelly, D., ‘Products of lattice varieties’, Algebra Universalis 21 (1985), 33–45.Google Scholar
[9]Hermann, Ch., ‘S-verklebte Summen von Verbänden’, Math. Z. 130 (1973), 225–274.CrossRefGoogle Scholar
[10]Ježek, J. and Slavik, A., ‘Primitive lattices’, Czechoslovak Math. J. 29 (104) (1979), 595–634.CrossRefGoogle Scholar
[11]Jónsson, B., ‘Universal relation systems’, Math. Scand. 4 (1956), 193–208.CrossRefGoogle Scholar
[12]Slavik, A., ‘A note on the amalgamation property in lattice varieties’, Comment. Math. Univ. Carolinae 21 (1980), 473–478.Google Scholar