Completeness, reducibility, primarity and purity for algebras: results and problems
The starting point of the motivation of the author is the structure theory of abelian groups from the point of view of varieties of groups. A typical example is the fact that an abelian group is divisible if and only if it does not have nontrivial homomorphic images in the atoms of the lattice of varieties of abelian groups (i.e., in the varieties of abelian groups of prime exponent). By analogy, if \(L({\mathcal V})\) is the lattice of subvarieties of a variety \(\mathcal V\) of algebraic systems, then the author calls the algebras in \(\mathcal V\) complete if they do not have nontrivial homomorphic images in the atoms of \(L({\mathcal V})\). An algebra is reduced if it does not contain nontrivial complete subalgebras. In the same sense, the author defines other analogues of structure properties from group theory. The article under review is a survey on the concepts of completeness, reducibility, primarity, and purity for arbitrary algebraic systems and for groups, modules, monoassociative algebras, semigroups, lattices and unars. A list of 28 main problems is given and it is shown how far the progress for different important algebraic systems is.
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