Algebras and groups defined by permutation relations of alternating type

The class of finitely presented algebras over a field $K$ with a set of generators $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n$ and defined by homogeneous relations of the form $a_1{a}_2\mathrm{.}\mathrm{.}\mathrm{.}{a}_n=a_{\sigma (1)}{a}_{\sigma (2)}\mathrm{.}\mathrm{.}\mathrm{.}{a}_{\sigma (n)}$, where $\sigma$ runs through ${\text{Alt}}_n$, the alternating group, is considered. The associated group, defined by the same (group) presentation, is described. A description of the radical of the algebra is found. It turns out that the radical is a finitely generated ideal that is nilpotent and it is determined by a congruence on the underlying monoid, defined by the same presentation.