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A Characterization of Sharply Transferable Lattices

Published online by Cambridge University Press:  20 November 2018

G. Grätzer  and
C. R. Platt
G. Grätzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba
C. R. Platt
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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A lattice L is called transferable if and only if, whenever L can be embedded in the ideal lattice I(K) of a lattice K, L can be embedded in K. L is called sharply transferable if and only if, for every lattice embedding ψ(x) , there exists an embedding such that for x, y ϵ L, if and only if x ≤ y. Finite sharply transferable lattices were characterized in [3]. In this paper we extend the characterization to the infinite case. We begin by revising some of the terminology of [3].

1.1. Definition, (a) Let 〈P; ≧〉 be a poset and X, Y ⊆ P. Then X dominates Y (written X Dom Y) if and only if, for every y ∈ F, there exists x ∈ X such that y ∈ x. Dually, X supports Y (written X Spp Y) if and only if, for every y ∈ Y, there exists x ∈ X such that x ≦ y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 145 - 154
Copyright
Copyright © Canadian Mathematical Society 1980

References

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