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A commutative doubly-idempotent semiring (cdi-semiring) (S,V,·,0,1) is a semilattice (S,V,0) with x V 0 = x and a semilattices (S,·,1) with identity 1 such that x0 = 0, and x(y V z) = xy V xz holds for all x, y, z ϵ S. Bounded distributive lattices are cdi-semirings that satisfy xy = x ^ y, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a V-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 nth Catalan number Cn = (1/(n+1)) (2n/n ) . We also show that cdi-semirings with a complete atomic Boolean V-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 cdisemirings of size n and all Boolean cdi-semirings of size 2n.
idempotent semirings, distributive lattices, preorder forests
N. Galataos and P. Jipsen, Commutative doubly-idempotent semirings determined by chains and by preorder forests, Relational and Algebraic Methods in Computer Science (Warsaw, 1991), Springer International Publ., vol. 18, RAMiCS, Palaiseau, 2020, pp. 1-14.