Reverse Lexicographic and Lexicographic Shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, ${\mathrm{\Delta }}_{\text{lex}}$ – an operation that transforms a monomial ideal of $S=\text{field}[x_i:i\in \mathbb{N}]$ that is finitely generated in each degree into a squarefree strongly stable ideal – is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal $I\subset S$ is fixed by lexicographic shifting if and only if $I$ is a universal squarefree lexsegment ideal (abbreviated USLI) of $S$. Moreover, in the case when $I$ is finitely generated and is not a USLI, it is verified that all the ideals in the sequence $\{{\mathrm{\Delta }}_{\text{lex}}^i(I){\}}_{i=0}^{\mathrm{\infty }}$ are distinct. The limit ideal $\bar{\mathrm{\Delta }}(I)={\lim }_{i\to \mathrm{\infty }}{\mathrm{\Delta }}_{\text{lex}}^i(I)$ is well defined and is a USLI that depends only on a certain analog of the Hilbert function of $I$.