  By using some lattice-like operations which constitute extensions of ones
introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new
perspective is gained on the subject of isotonicity of the metric projection
onto the closed convex sets. The results of this paper are wide range
generalizations of some results of the authors obtained for self-dual cones.
The aim of the subsequent investigations is to put into evidence some closed
convex sets for which the metric projection is isotonic with respect the order
relation which give rise to the above mentioned lattice-like operations. 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 and respectively by H. Nishimura and E.
A. Ok.
