  The catalytic reaction system (CRS) formalism by Hordijk and Steel is a
versatile method to model autocatalytic biochemical reaction networks. It is
particularly suited, and has been widely used, to study self-sustainment and
self-generation properties. Its distinguishing feature is the explicit
assignment of a catalytic function to chemicals that are part of the system. In
this work, it is shown that the subsequent and simultaneous catalytic functions
gives rise to an algebraic structure of a semigroup with the additional
compatible operation of idempotent addition and a partial order. The aim of
this article is to demonstrate that such semigroup models are a natural setup
to describe and analyze self-sustaining CRS. The basic algebraic properties of
the models are established and the notion of the function of any set of
chemicals on the whole CRS is made precise. This leads to a natural discrete
dynamics on the network, which results from iteratively considering the
self-action on a set of chemicals by its own function. The fixed points of this
dynamics are proven to correspond to self-sustaining sets of chemicals, which
are functionally closed. Finally, as the main application, a theorem on the
maximal self-sustaining set and a structure theorem on the lattice of
functionally closed self-sustaining sets of chemicals are proven.
