GENERALIZED DERIVED ALGEBRAS AND GENERALIZED INDUCED ALGEBRAS

Substituting for the fundamental operations of an algebra term operations we get a new algebra of the same type, called a generalized derived algebra. Such substitutions are called generalized hypersubstitutions. Generalized hypersubstitutions can also be applied to every equation of a fully invariant equational theory. The equational theory generated by the resulting set of the equations induces on every algebra of the type under consideration a fully invariant congruence relation. If we factorize the generalized derived algebra by this fully invariant congruence relation we will obtain an algebra which we call generalized induced algebra. In this paper, we prove some properties which transfer the starting algebras to generalized derived algebras and to generalized induced algebras.

EQUATIONAL THEORIES GENERATED BY HYPERSUBSTITUTIONS OF TYPE (n)

Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitutions are fully invariant congruence relations on the (absolutely free) term algebra of the considered type. For an arbitrary type τ = (n), n ≥ 1, i.e. if one has only one n-ary operation symbol, we will describe all these congruence relations. The results will be applied to solve the hyperunification problem. Further we will give some generalizations to arbitrary types.

On equivalence of semigroup identities

For a given relation $\rho$ on a free semigroup ${\mathcal F}$ we describe the smallest cancellative fully invariant congruence ${\rho}^{\sharp}$ containing $\rho$. Two semigroup identities are s-equivalent if each of them is a consequence of the other on cancellative semigroups. If two semigroup identities are equivalent on groups, it is not known if they are s-equivalent. We give a positive answer to this question for all binary semigroup identities of the degree less or equal to 5. A poset of corresponding varieties of groups is given.