Narrow Results

In this paper, the left (right) $h$-nucleus of an invertible algebra is defined and its connection with regular permutations of the invertible algebra is investigated. Using the notions of the $h$-nucleus, we have obtained the characterizations of linear invertible algebras and those for left (right) linear invertible algebras by the second-order formulas.

In this paper, using the second-order formulas, we obtained characterizations of invertible algebras principally isotopic to a group or an abelian group.

In this paper, the Belousov theorem on linearity of invertible algebras with the Schauffler $\forall \exists (\forall )$-identity is extended over the other $\forall \exists (\forall )$-identities of associativity. As a consequence, we obtain the equivalency of the considered $\forall \exists (\forall )$-identities of associativity and nontrivial hyperidentities of associativity in systems of groups. For the considered formulas, we prove the Schauffler-type theorems, too.