Geodesic
Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups
Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 395-397 Cet article a éte moissonné depuis la source Math-Net.Ru

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Two groups are constructed, these being elementarily equivalent, with one of them being nontrivially partially ordered but without being directionally ordered, while the other is directionally ordered. This proves the elementary nonclosure and nonaxiomatizability of the class of directionally ordered groups in the class of nontrivially partially ordered groups. It is shown that, in the decreasing chain of classes, i.e., all groups, nontrivially partially ordered groups, directionally ordered groups, and lattice ordered groups, each successive class is not elementarily closed, and hence, is not axiomatizable, in any earlier class. 
@article{MZM_1973_14_3_a9,
     author = {A. A. Vinogradov},
     title = {Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {395--397},
     year = {1973},
     volume = {14},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_3_a9/}
}
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A. A. Vinogradov. Nonaxiomatizability of directionally ordered groups in the class of nontrivially partially ordered groups. Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 395-397. http://geodesic.mathdoc.fr/item/MZM_1973_14_3_a9/