Two embedding theorems for lattices
HTML articles powered by AMS MathViewer
- by G. Grätzer and C. R. Platt
- Proc. Amer. Math. Soc. 69 (1978), 21-24
- DOI: https://doi.org/10.1090/S0002-9939-1978-0465963-5
- PDF | Request permission
Abstract:
A lattice L satisfies $(S{D_ \wedge })$ if $a \wedge b = a \wedge c$ implies that $a \wedge b = a \wedge (b \vee c) ((\mathrm {SD}_\vee )$ is defined dually). Theorem. Every lattice can be embedded in the ideal lattice of a lattice satisfying $(\mathrm {SD}_\wedge )$ (respectively, $(\mathrm {SD}_\vee )$). Call a lattice K transferable iff whenever K can be embedded in the ideal lattice of a lattice L, then K can be embedded in L. Corollary. Every transferable lattice satisfies $(\mathrm {SD}_\wedge )$ and $(\mathrm {SD}_\vee )$.References
- Rachad Antonius and Ivan Rival, A note on Whitman’s property for free lattices, Algebra Universalis 4 (1974), 271–272. MR 357251, DOI 10.1007/BF02485736
- Kirby A. Baker and Alfred W. Hales, From a lattice to its ideal lattice, Algebra Universalis 4 (1974), 250–258. MR 364036, DOI 10.1007/BF02485732
- H. S. Gaskill and C. R. Platt, Sharp transferability and finite sublattices of free lattices, Canadian J. Math. 27 (1975), no. 5, 1036–1041. MR 389687, DOI 10.4153/CJM-1975-109-7
- J. C. Abbott (ed.), Trends in lattice theory, Van Nostrand Reinhold Mathematical Studies, No. 31, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. Symposium held at the United States Naval Academy in May of 1966. MR 0270980
- G. Grätzer, A property of transferable lattices, Proc. Amer. Math. Soc. 43 (1974), 269–271. MR 335378, DOI 10.1090/S0002-9939-1974-0335378-5 —, General lattice theory, Birkhäuser, Basel, 1978.
- Philip M. Whitman, Free lattices, Ann. of Math. (2) 42 (1941), 325–330. MR 3614, DOI 10.2307/1969001
- Philip M. Whitman, Lattices, equivalence relations, and subgroups, Bull. Amer. Math. Soc. 52 (1946), 507–522. MR 16750, DOI 10.1090/S0002-9904-1946-08602-4
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 69 (1978), 21-24
- MSC: Primary 06A35
- DOI: https://doi.org/10.1090/S0002-9939-1978-0465963-5
- MathSciNet review: 0465963