Quiver algebras from semirings and posets as semirings A semiring is a nonempty set $S$ such two binary operations + and * making S into a semigroup with + and * and such that a*(b+c)=ab+ac and (b+c)a=ba+c*a for all a,b,c in S.
Assume in the following that all semirings are finite.
For example distributive lattices are semirings with join and meet and one can associate to every distributive lattice in a natural way a connected quiver algebra, namely the incidence algebra of the distributive lattice.

Question 1: Is there a natural way to associate a connected quiver algebra to a more genreal class of semirings that generalised the incidence algebras for distributive lattices?


Question 2: Are there other classes or even a classification of posets that have a natural semiring structure?

 A: Residuated lattices and some more general constructions have natural semiring structure.
A binary operation $\cdot$ on a poset $X$ is said to be residuated if there are binary operations $/,\backslash$ where for each $x,y,z\in X$, the following are equivalent:

*

*$x\cdot y\leq z$.


*$x\leq z/y$.


*$y\leq x\backslash z.$
Proposition: Suppose that $\cdot$ is residuated. If $R\subseteq X$ and $\bigvee R$ exists, then
$x\cdot\bigvee R=\bigvee_{r\in R}(x\cdot r)$ and
$(\bigvee R)\cdot y=\bigvee_{r\in R}(r\cdot y)$.
Proof: $x\cdot\bigvee R\leq z$ if and only if $\bigvee R\leq x\backslash z$ if and only if
$\forall r\in R,r\leq x\backslash z$ if and only if
$\forall r\in R,x\cdot r\leq z$. Therefore, $x\cdot\bigvee R=\bigvee_{r\in R}(x\cdot r)$.
The proof that $(\bigvee R)\cdot y=\bigvee_{r\in R}(r\cdot y)$ is similar. $\square$
There are plenty of examples of residuated lattices.
Example 0: The collection of all ideals in a ring is a residuated lattice.
Example 1: Every Heyting algebra $(X,\wedge,\vee,\rightarrow,1)$ is a residuated lattice where we define $a\backslash b=b/a=a\rightarrow b$. But then again, in a Heyting algebra, the resulting semiring is the semiring structure on distributive lattices mentioned in the question.
Example 2: If $(G,\cdot)$ is a group that is partially ordered in the sense that if $r,s,x,y\in G$, then $x\leq y$ implies that $r\cdot x\leq r\cdot y$ and $x\cdot s\leq y\cdot s$. Then the binary operation $\cdot$ on $G$ is residuated.
Example 3: If $X$ is a set, then $P(X\times X)$ is partially ordered by $\subseteq$, but $P(X\times X)$ is also a monoid under the composition of relations. The operation $\circ$ is also residuated. It is very easy to directly show that $R\circ(S\cup T)=(R\circ S)\cup(R\circ T)$ and $(R\cup S)\circ T=(R\circ T)\cup(S\circ T)$.
