  M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and
constructed the invariant P of framed links in 3-space. The invariant is
closely related to the Kauffman bracket polynomial. The normalized bracket
polynomial is obtained from the Kauffman bracket polynomial by the
multiplication of indeterminate and it is an ambient isotopy invariant for
links. In this paper, we reformulate the multiplication by using a map from the
set of framed links to a Kauffman bracket magma in order that $P$ is invariant
for links in 3-space. We define a generalization of a Kauffman bracket magma,
which is called a marked Kauffman bracket magma. We find the conditions to be
invariant under Yoshikawa moves except the first one and use a map from the set
of admissible marked graph diagrams to a marked Kauffman bracket magma to
obtain the invariant for surface-links in 4-space.
