Unified products for alternative and pre-alternative algebras.

In this paper, the unified products for alternative and pre-alternative algebras is studied. It is shown that there exists an alternative algebra structure on $E$ such that $A$ is a subalgebra of $E$ if and only if $E$ is isomorphic to an extending structures of $A$ and $V$. Two cohomological type objects $\mathcal{H}_A^2(V,A)$ and $\mathcal{H}^2(V,A)$ are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension $A\subset E$ of alternative algebras, another cohomological type object is constructed to classify all complements of $A$ in $E$.