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Home > Vol 23, No 2 (2006)

Eric Babson , Isabella Novik and Rekha Thomas

University of Washington Department of Mathematics Seattle WA 98195-4350 Seattle WA 98195-4350

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ  _lex-an operation that transforms a monomial ideal of S = K[ x _i: i \in  \Bbb 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 { D _lex ⁱ ( I) } _i=0 ^\? \{ \Delta_{\rm lex}^{i} (I) \}_{i=0}^{\infty} } are distinct. The limit ideal [ `( D)]( I) = lim _{i \textregistered  \?} D _lex ⁱ ( I) \bar{Δ}(I) = {\rm lim}_{i \rightarrow \infty} \Delta_{\rm lex}^{i} (I) is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I.

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