A first order structure with universe M is atomic compact if every system of atomic formulas with parameters in M is satisfiable in provided each of its finite subsystems is. We consider atomic compactness for the class of reflexive (symmetric) graphs. In particular, we investigate the extent to which “sparse” graphs (i.e. graphs with “few” vertices of “high” degree) are compact with respect to systems of atomic formulas with “few” unknowns, on the one hand, and are pure restrictions of their Stone-Čech compactifications, on the other hand.
@article{bwmeta1.element.bwnjournal-article-fmv162i2p99bwm,
author = {Christian Delhomm\'e},
title = {Atomic compactness for reflexive graphs},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {99-117},
zbl = {0946.03043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv162i2p99bwm}
}
Delhommé, Christian. Atomic compactness for reflexive graphs. Fundamenta Mathematicae, Tome 159 (1999) pp. 99-117. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv162i2p99bwm/
[00000] [1] E. Corominas, Sur les ensembles ordonnés projectifs et la propriété du point fixe, C. R. Acad. Sci. Paris Sér. I 311 (1990), 199-204. | Zbl 0731.06001
[00001] [2] C. Delhommé, Propriétés de projection, thèse, Université Claude Bernard-Lyon I, 1995.
[00002] [3] C. Delhommé, Infinite projection properties, Math. Logic Quart. 44 (1998), 481-492. | Zbl 0922.03049
[00003] [4] S. Hazan, On triangle-free projective graphs, Algebra Universalis 35 (1996), 185-196. | Zbl 0845.05062
[00004] [5] R. McKenzie and S. Shelah, The cardinals of simple models for universal theories, in: Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1971, 53-74.
[00005] [6] J. Mycielski, Some compactifications of general algebras, Colloq. Math. 13 (1964), 1-9. | Zbl 0136.26102
[00006] [7] J. Mycielski and C. Ryll-Nardzewski, Equationally compact algebras II, Fund. Math. 61 (1968), 271-281.
[00007] [8] W. Taylor, Atomic compactness and graph theory, ibid. 65 (1969), 139-145. | Zbl 0182.34401
[00008] [9] W. Taylor, Compactness and chromatic number, ibid. 67 (1970), 147-153. | Zbl 0199.30401
[00009] [10] W. Taylor, Some constructions of compact algebras, Ann. Math. Logic 3 (1971), 395-435. | Zbl 0239.08003
[00010] [11] W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33-53. | Zbl 0263.08005
[00011] [12] B. Węglorz, Equationally compact algebras (I), Fund. Math. 59 (1966), 289-298. | Zbl 0221.02039
[00012] [13] F. Wehrung, Equational compactness of bi-frames and projection algebras, Algebra Universalis 33 (1995), 478-515. | Zbl 0830.08005
[00013] [14] G. Wenzel, Equational compactness, Appendix 6 in: G. Grätzer, Universal Algebra, 2nd ed., Springer, 1979.