# tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal

## Synopsis

• Usage:
tStronglyStableIdeal(I,t)
• Inputs:
• I, a t-spread ideal of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• a list, the smallest t-strongly stable ideal containing the t-spread ideal I

## Description

the function tStronglyStableIdeal(I,t) gives the smallest t-strongly stable ideal containing I, that is, $B_t(G(I))$ where $G(I)$ is the minimal set of generators of $I$
We recall that if $u\in M_{n,d,t}\subset S=K[x_1,\ldots,x_n]$ then $B_t(u)$ is the smallest t-strongly stable ideal of $S$ containing $u.$
Moreover, a subset $N\subset M_{n,d,t}$ is called a t-strongly stable set if taking a t-spread monomial $u\in N$, for all $j\in \mathrm{supp}(u)$ and all $i,\ 1\leq i\leq j$, such that $x_i(u/x_j)$ is a t-spread monomial, then it follows that $x_i(u/x_j)\in N$.
A t-spread monomial ideal I is t-strongly stable if $[I_j]_t$ is a t-strongly stable set for all $j$, where $[I_j]_t$ is the t-spread part of the $j$-th graded component of I.

Examples:

 i1 : S=QQ[x_1..x_9] o1 = S o1 : PolynomialRing i2 : tStronglyStableIdeal(ideal {x_2*x_5*x_8},2) o2 = ideal (x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , 1 3 5 1 3 6 1 3 7 1 4 6 1 3 8 1 4 7 2 4 6 1 4 8 ------------------------------------------------------------------------ x x x , x x x , x x x , x x x , x x x , x x x ) 1 5 7 2 4 7 1 5 8 2 4 8 2 5 7 2 5 8 o2 : Ideal of S i3 : tStronglyStableIdeal(ideal {x_2*x_5*x_8},3) o3 = ideal (x x x , x x x , x x x , x x x ) 1 4 7 1 4 8 1 5 8 2 5 8 o3 : Ideal of S