Lattice-like subsets of Euclidean Jordan Algebras

While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder, and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper, it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by methods for proving the existence of solutions of variational inequalities and methods for finding these solutions in a recursive way. It turns out that the closed convex sets admitting isotone projections are exactly the sets which are invariant with respect to these lattice-like operations, called lattice-like sets. In this paper, it is shown that the Jordan subalgebras are lattice-like sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least 3, the lattice-like property is rather restrictive, e.g., there are no lattice-like proper closed convex sets with interior points.