  The algebraic K-theory of Lawvere theories is a conceptual device to
elucidate the stable homology of the symmetry groups of algebraic structures
such as the permutation groups and the automorphism groups of free groups. In
this paper, we fully address the question of how Morita equivalence classes of
Lawvere theories interact with algebraic K-theory. On the one hand, we show
that the higher algebraic K-theory is invariant under passage to matrix
theories. On the other hand, we show that the higher algebraic K-theory is not
fully Morita invariant because of the behavior of idempotents in non-additive
contexts: We compute the K-theory of all Lawvere theories Morita equivalent to
the theory of Boolean algebras.
