  It is well known from universal algebra that, for every signature $\Sigma$,
there exist algebras over $\Sigma$ which are absolutely free, meaning that they
do not satisfy any identities or, alternatively, satisfy the universal mapping
property for the class of $\Sigma$-algebras. Furthermore, once we fix a
cardinality of the generating set, they are, up to isomorphisms, unique, and
equal to algebras of terms (or propositional formulas, in the context of
logic). Equivalently, the forgetful functor, from the category of
$\Sigma$-algebras to $\textbf{Set}$, has a left adjoint. This result does not
extend to multialgebras. Not only multialgebras satisfying the universal
mapping property do not exist, but the forgetful functor $\mathcal{U}$, from
the category of $\Sigma$-multialgebras to $\textbf{Set}$, does not have a left
adjoint.
  In this paper we generalize, in a natural way, algebras of terms to
multialgebras of terms, whose family of submultialgebras enjoys many properties
of the former. One example is that, to every pair consisting of a function,
from a multialgebra of terms to another multialgebra, and a collection of
choices (which selects how a homomorphism approaches indeterminacies), it
corresponds a unique homomorphism, which ressembles the universal mapping
property. Another example is that the multialgebras of terms are generated by a
set that may be viewed as a strong basis, which we call the ground of the
multialgebra. Submultialgebras of multialgebras of terms are what we call
weakly free multialgebras. Finally, with these definitions at hand, we offer a
simple proof that multialgebras with the universal mapping property for the
class of all multialgebras do not exist and that $\mathcal{U}$ does not have a
left adjoint.
