Narrow Results

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.

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set W_τ(X) of all terms of type τ. If σ : {f_i | i ∈ I} → W_τ(X) is a hypersubstitution and its canonical extension, then the set

Many-sorted algebras are used in Computer Science for abstract data type specifications. It is widely believed that many-sorted algebras are the appropriate mathematical tools to explain what abstract data types are ([1]). In this paper we want to extend the concept of a hypersubstitution from the one-sorted to the many-sorted case. Hypersubstitutions are used for one-sorted algebras to define hyperidentities and M-solid varieties ([2]). We will prove that extensions of hypersubstitutions for many-sorted algebras are endomorphisms of the many-sorted clone of a given type. As in the one-sorted case we define a binary operation for hypersubstitutions and prove that with respect to this operation all many-sorted hypersubstitutions form a monoid.

We determine all V-proper hypersubstitutions for the variety of pseudo-complemented semilattices. Moreover, for any monoid of hypersubstitutions we will give the greatest M-solid variety of pseudo-complemented semilattices.

Hypersubstitutions map operation symbols to terms of the corresponding arities. Using hypersubstitutions from each algebra one can derive new algebras. We compare properties of congruence lattices, of subalgebra lattices and of clones of the derived algebras with the corresponding properties of the starting algebras. If both algebras have the same clone of term operations, then they have the most properties in common. This is for instance the case if the hypersubstitution is surjective. We characterize surjective hypersubstitutions and prove that for finite types surjective hypersubstitutions are bijective. This will be applied to i-closed varieties of n-ary type.