Let $k$ be a field and let $R$ be the polynomial ring $k[x_1,\ldots,x_n]$. Let $I$ be a monomial ideal of $R$. We say that $I$ is stable if it satisfies the following "exchange property": for every monomial $m\in I$ and index $i<max(m)$, the monomial $x_im/x_{max(m)}\in I$, where $max(m)$ is the largest index of a variable dividing $m$. Basically, you can exchange the variable with the largest index appearing in $m$ with something with smaller index and still remain in the ideal $I$.
It is known that there is a unique function $g:M(I)\rightarrow G(I)$, where $M(I)$ is the set of monomials in $I$ and $G(I)$ is the set of minimal generators of $I$, such that $m=g(m)c(m)$ for some $c(m)$ with $max(g(m))\leq min(c(m))$, where min is defined similarly as to the max. Roughly speaking, you want to order the variables of $m$ as $x_1^{a_1}\cdots x_n^{a_n}$, and then start scanning $m$ from left to right until you hit a minimal generator of $I$, and call that generator $g(m)$.
I was reading the paper of Peeva on the existence of a DG algebra structure on the Eliahou-Kervaire resolution, you can find that paper here. In this paper, she takes two minimal generators of $I$, say $a$ and $b$, and defines the $I$-meet of $a$ and $b$ to be the element $g(lcm(a,b))$. Because of the terminology she uses, it makes me think there is a lattice hiding behind the scenes. I know about the lcm-lattice of a monomial ideal, and I do not think it is that one.
So my question is, do you see a lattice that would justify the terminology that Peeva is using in this paper?