Monomial Ideals under Ideal Operations

Abstract

In this article, we show for a monomial ideal I of K[x 1, x 2,…, x n ] that the integral closure is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if I has the same property. We also show that the kth symbolic power I (k) of I preserves the properties of Borel type, Borel-fixed and strongly stable, and I (k) is lexsegment if I is stably lexsegment. For a monomial ideal I and a monomial prime ideal P, a new ideal J(I, P) is studied, which also gives a clear description of the primary decomposition of I (k). Then a new simplicial complex J Δ of a monomial ideal J is defined, and it is shown that . Finally, we show under an additional weak assumption that a monomial ideal is universal lexsegment if and only if its polarization is a squarefree strongly stable ideal.