In this project we will try to describe the commutative doubly-idempotent semiring (cdi-semiring). A cdi-semiring (S, \/, ∙, 0, 1) is a semilattice (S, \/, 0) with x \/ 0 = 0 and a semilattices (S, ∙, 1) with identity 1 such that x0 = 0, and x(y \/ z) = xy \/ xz holds for all x,y,z in S. Bounded distributive lattices are cdi-semirings that satisfy xy = x /\ y, and the variety of cdi-semirings covers the variety of distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by the 3-element cdi-semiring.
We show that there are seven cdi-semirings with a \/-semilattice of height less than or equal to 2. We construct all cdi-semirings for which their multiplicative semilattice is a chain with n+1 elements, and we show that up to isomorphism the number of such algebras is the n-th Catalan number.
We also show that cdi-semirings with a complete atomic Boolean \/-semilattice on the set of atoms A are determined by singleton-rooted preorder forests on the set A. From these results we obtain efficient algorithms to construct all multiplicatively linear cdi-semirings of size n and all Boolean cdi-semirings of size 2^n.