Geodesic
Injectives and Projectives in Term Finite Varieties of Algebras
Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 769-775

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Let V be a class of similar algebras. An algebra is V-injective provided and whenever and fis a one-to-one homomorphism from into and g is a homomorphism from into , then there is a homomorphism h from into such that h o f = g. So is injective provided all diagrams of the following sort can be completed. Dually, is V-projective provided and whenever and f is a homomorphism from onto and g is a homomorphism from into , then there is a homomorphism h from into such that f o h = g. So is projective provided all diagrams of the following sort can be completed: This usage of the words “projective” and “injective” differs somewhat from the usage current in category theory. 
McNulty, George F.; Nordahl, T.; Scheiblich, H. E. Injectives and Projectives in Term Finite Varieties of Algebras. Canadian journal of mathematics, Tome 35 (1983) no. 5, pp. 769-775. doi: 10.4153/CJM-1983-044-5
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