  In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$
that the integral closure $\ol{I}$ 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 $k^{th}$
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 $_J\bigtriangleup$ of
a monomial ideal $J$ is defined, and it is shown that
$I_{_J\bigtriangleup^{\vee}} = \sqrt{J}$. 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.
