Geodesic
Graphical Compositions and Weak Congruences
Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 34

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Graphical compositions of equivalences were introduced (independently) by B. Jónsson and H. Werner in order to determine whether a subset of Eq$(X)$ (the set of all equivalences on the set $X$) is the set of all congruences of some algebra defined on $X$. Namely, a complete sublattice $L$ of Eq$(X)$ is the congruence lattice of some algebra defined on $X$ if and only if $L$ is closed under all graphical compositions. We generalize this result and prove that a similar characterization is possible for weak congruences (i.e., symmetric and transitive compatible relations).
Classification : 03A30 08A40 
@article{PIM_1994_N_S_56_70_a4,
     author = {Miroslav Plo\v{s}\v{c}ica},
     title = {Graphical {Compositions} and {Weak} {Congruences}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {34 },
     publisher = {mathdoc},
     volume = {_N_S_56},
     number = {70},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1994_N_S_56_70_a4/}
}
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Miroslav Ploščica. Graphical Compositions and Weak Congruences. Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 34 . http://geodesic.mathdoc.fr/item/PIM_1994_N_S_56_70_a4/