Consider the following construction : let $(L,\vee,\wedge)$ be a finite distributive lattice, and let $(\mathrm{Int}(L),\star)$ be the monoid defined on the set of non empty intervals of $L$ $$\mathrm{Int}(L)=\lbrace[a,b]\mid a,b\in L, a\leq b\rbrace$$ with multiplication given by the formula $$ [a,b]\star[c,d]=[(a\vee c)\wedge b,(a\vee d)\wedge b] $$ (associativity of this multiplication turns out to be equivalent to distributivity of $L$ or equivalently to the requirement that the composite of projections of the sort $\pi^+_{[a,b]}:L\to L,x\mapsto (a\vee x)\wedge b$ is again a projection). This monoid turns out to be a left regular band (LRB) which means that for all intervals $I$, $J$ one has $I\star I=I$ and $I\star J\star I=I\star J$. The unit is $L=[0_L,1_L]$.

To any (finite unital) LRB $(B,\cdot)$ one associates its intersection lattice $(\Lambda_B,\cap)$ which is the lattice of all principal left ideals $Bx$ (one has $Bxy=Bx\cap By$ so that $B\to\Lambda_B,x\mapsto Bx$ is a morphism of lattices; details may be found in the paper Poset topology and homological invariants of algebras arising in algebraic combinatorics by S. Margolis, F. Saliola and B. Steinberg.) In the case of the monoid of intervals of a distributive lattice, I'll write $\Lambda_L=\Lambda_{\mathrm{Int}(L)}$.

The right Green order is given by inclusion of intervals. It's possible to characterize the intervals $I$, $J$ that define the same principal left ideal : $\mathrm{Int}(L)I=\mathrm{Int}(L)J$ iff $I$ and $J$ are equivalent under the equivalence relation of perspective (projectivity?), which is the equivalence relation generated by $[a\wedge b,a]\sim[b,a\vee b]$. One proves that $\Lambda_L$ is **atomistic lower-semimodular**. It is easy to see that the $\mathscr{L}$-classes of $\Lambda_L$ (i.e. the sets of intervals that generate the same principal left ideal) form distributive lattices themselves.

Atomistic

upper-semimodular lattices are known as geometric lattices and are of interest in matroid theory : quoting from Wikipedia "geometric lattices [...] form the lattices of flats of finite [...] matroids".1) Do atomistic

lower-semimodular lattices have any interpretation or relevance?2) Also : is this simple construction already known?

I plan to include this family of examples in my thesis, and would like to properly reference them if it already exists somewhere.