# isTStronglyStableSeg -- whether the given list of t-spread monomials is a t-strongly stable segment

## Synopsis

• Usage:
isTStronglyStableSeg(l,t)
• Inputs:
• l, a list of t-spread monomials of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• , whether the list l of t-spread monomials is a t-strongly stable segment

## Description

the function isTStronglyStableSeg(l,t) is true whether the list of monomials l is a t-strongly stable segment, that is, the set of all the t-spread monomials belonging to the strongly stable set generated by the smallest monomial of l and smaller than the greatest monomial of l, with respect to $>_\mathrm{slex}.$
In other words, whether $l=B_t[\max l,\min l]=\{w\in B_t\{\min l\}\ :\ \max l\geq_\mathrm{slex} w\}.$
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 set of monomials of $M_{n,d,t}$ 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 $x_i(u/x_j)\in N$.

Examples:

 i1 : S=QQ[x_1..x_9] o1 = S o1 : PolynomialRing i2 : isTStronglyStableSeg({x_1*x_4*x_7,x_1*x_4*x_8,x_1*x_5*x_8,x_2*x_5*x_8},3) o2 = true i3 : isTStronglyStableSeg({x_1*x_4*x_7,x_1*x_4*x_8,x_2*x_5*x_8},3) o3 = false