Jets and principal components of monomial ideals and very well-covered graphs

Motivated by using combinatorics to study jets of monomial ideals, we extend a definition of jets from graphs to clutters. We offer some structural results on their vertex covers and show an interesting connection between the cover ideal of the jets of a clutter and the symbolic powers of the cover ideal of the original clutter. We use this connection to prove that jets of very well-covered graphs are very well covered. Next, we turn our attention to principal jets of monomial ideals, describing their primary decomposition and minimal generating sets. Finally, we give formulas to compute various algebraic invariants of principal jets of monomial ideals, including their Hilbert series, Betti numbers, multiplicity, and regularity.