Commutative Doubly-Idempotent Semirings
Published research
Research Summary
A commutative doubly-idempotent semiring has idempotent addition and multiplication. Its additive join operation has identity 0, so x ∨ 0 = x; multiplication has identity 1, distributes over joins, and satisfies x · 0 = 0.
We describe these structures when the multiplicative semilattice is a chain, showing that their isomorphism classes are counted by Catalan numbers. We also connect Boolean examples with singleton-rooted preorder forests, leading to algorithms for constructing finite semirings.
Publication Details
- N. Alpay and P. Jipsen. Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests. In Relational and Algebraic Methods in Computer Science (RAMiCS 2020), Lecture Notes in Computer Science 12062, Springer, 2020, pp. 1–14.