Semiring identities of semigroups of reflexive relations and upper triangular boolean matrices

TL;DR

This paper demonstrates that certain semirings related to reflexive relations, upper triangular boolean matrices, and order-preserving transformations satisfy the same identities, and that these identities have finite bases.

Contribution

It establishes the equivalence of identities among these semirings and proves the finiteness of their identity bases, extending previous results.

Findings

Semirings $ ext{R}_n$, $ ext{U}_n$, and $ ext{C}_n$ satisfy the same identities.

The identities of $ ext{R}_n$ and $ ext{U}_n$ have finite bases.

Extension of finite basis results to new classes of semirings.

Abstract

We show that the following semirings satisfy the same identities: the semiring $\mathcal{R}_n$ of all reflexive binary relations on a set with $n$ elements, the semiring $\mathcal{U}_n$ of all $n\times n$ upper triangular matrices over the boolean semiring, the semiring $\mathcal{C}_n$ of all order preserving and extensive transformations of a chain with $n$ elements. In view of the result of Kl\'ima and Pol\'ak, which states that $\mathcal{C}_n$ has a finite basis of identities for all $n$, this implies that the identities of $\mathcal{R}_n$ and $\mathcal{U}_n$ admit a finite basis as well.