A groupoid $H$ is a homomorphic image of a subdirectly irreducible groupoid $G$ (over its monolith) if and only if $H$ has a smallest ideal.
@article{119258, author = {David Stanovsk\'y}, title = {Homomorphic images of subdirectly irreducible groupoids}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {42}, year = {2001}, pages = {443-450}, zbl = {1057.20049}, mrnumber = {1859591}, language = {en}, url = {http://dml.mathdoc.fr/item/119258} }
Stanovský, David. Homomorphic images of subdirectly irreducible groupoids. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 443-450. http://gdmltest.u-ga.fr/item/119258/
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