  By using the metric projection onto a closed self-dual cone of the Euclidean
space, M. S. Gowda, R. Sznajder and J. Tao have defined generalized lattice
operations, which in the particular case of the nonnegative orthant of a
Cartesian reference system reduce to the lattice operations of the
coordinate-wise ordering. The aim of the present note is twofold: to give a
geometric characterization of the closed convex sets which are invariant with
respect to these operations, and to relate this invariance property to the
isotonicity of the metric projection onto these sets. As concrete examples the
Lorentz cone and the nonnegative orthant are considered. Old and recent results
on closed convex Euclidean sublattices due to D. M. Topkis, A. F. Veinott and
to M. Queyranne and F. Tardella, respectively are obtained as particular cases.
The topic is related to variational inequalities where the isotonicity of the
metric projection is an important technical tool. For Euclidean sublattices
this approach was considered by G. Isac, H. Nishimura and E. A. Ok.
