Construction of existentially closed Abelian lattice-ordered groups using Fraïssé limits

2021 ◽  
Vol 82 (1) ◽  
Author(s):  
Brian Wynne
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Existentially Closed ◽  
Abelian Lattice

Cevian properties in ideal lattices of Abelian ℓ-groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Miroslav Ploščica
Keyword(s):  
Distributive Lattice ◽  
Recent Result ◽  
Ideal Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Spectral Spaces ◽  
Abelian Lattice

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

scholarly journals Free abelian lattice-ordered groups

2005 ◽  
Vol 134 (2-3) ◽  
pp. 265-283 ◽  
Author(s):  
A.M.W. Glass ◽  
Angus Macintyre ◽  
Françoise Point
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice
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