  Any Lie group G acting on a Euclidean nonvoid open subset M can be seen as a
subgroup of the smooth diffeomorphisms Diff^\infty(M,M) of M into itself. Thus
actions by such Lie groups G correspond to smooth coordinate transforms on M
which, in particular, have smooth inverses. In Rosinger [1, chap. 13], the
study of Lie semigroups G in the vastly larger semigroup C^\infty(M,M) of
smooth maps of M into itself was initiated. Such semigroups were named "genuine
Lie semigroups", or in short, GLS, since they are no longer contained in
Diff^\infty(M,M), thus they correspond to smooth coordinate transforms which
need not have smooth inverses. Genuine Lie semigroups, or GLS, have a major
interest since they still can transform solutions of linear or nonlinear PDEs
into other solutions of the respective equations, thus leading to the vastly
larger class of "semi-symmetries" of such equations. Certain Lie semigroups
have earlier been studied in the literature, Hilgert, et.al. However, such
semigroups have always been contained in suitable Lie groups, thus they have
been contained in Diff^\infty(M,M) as sub-semigroups. In particular, the
coordinate transforms defined by them were always invertible, unlike in the
case of "genuine Lie semigroups", or GLS, studied here.
