Special Embeddings of Symmetric Varieties and their Borel orbits

We determine when an antiinvolution on a semisimple linear algebraic group $G$ extends to an antiinvolution on a J-irreducible monoid $M$ whose group of invertible elements is $G$. Using this information, we study a special class of compactifications of symmetric varieties first considered by Thierry Vust. Generalizing the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings.