  We initiate an algebraic approach to study DNA origami structures by
associating an element from a monoid to each structure. We identify two types
of basic building blocks and describe an DNA origami structure with their
composition. These building blocks are taken as generators of a monoid, called
origami monoid, and, motivated by the well studied Temperley-Lieb algebras, we
identify a set of relations that characterize the origami monoid. We also
present several observations about the Green's relations for the origami monoid
and study the relations to a cross product of Jones monoids that is a morphic
image of an origami monoid.
