Abstract
This chapter describes the endomorphically complete groups. Given the algebra A of a variety, one can ask whether every endomorphism of A is a polynomial function of A. It is found that if this is the case, one call A endomorphically complete. It is clear that any polynomially complete algebra is also endomorphically complete. It is observed that whereas any polynomially complete algebra is simple, there are endomorphically complete algebras with nontrivial congruence lattices. The congruence relations of such algebra are fully invariant. Subalgebras and homomorphic images of endomorphically complete algebras are rarely again endomorphically complete. For some varieties, there is a condition under which endomorphical completeness is preserved under direct products. The variety of groups is one of these and one can show that the direct product of G1, and G2 with orders | G1 | and | G2 | is again endomorphically complete if one have the greatest common divisor = 1.
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