Identities of triangular Boolean matrices

TL;DR

The paper characterizes semiring and semigroup identities of the ai-semi-ring of Boolean upper triangular $n\times n$ matrices, $T_n$, via a precise combinatorial criterion on subwords with gaps: for a word $\mathbf{u}$ of length $k<n$, any occurrence in the lower side of an inequality with gaps $G_1,\dots,G_{k+1}$ must correspond to an occurrence in the upper side with gaps $G'_1,\dots,G'_{k+1}$ satisfying $G'_\ell\subseteq G_\ell$ for all $\ell$. It further refines this in the semigroup case, provides a simplified, inductive criterion, and derives several key applications: a polynomial-time algorithm for identity checking in $T_n$; results on the Finite Basis Problem showing inherent non-finite baseness for $n\ge3$ (with partial results for ai-semi-rings); and a language-theoretic perspective linking identities to recognizable languages via pseudovarieties generated by $T_n$. These contributions illuminate the complexity of identity checking, the axiomatizability of equational theories, and algebraic descriptions of certain recognizable languages, advancing both algebraic and formal-language perspectives on triangular Boolean matrices.

Abstract

We give a combinatorial characterization of the identities holding in the semiring of all upper triangular Boolean $n\times n$-matrices and apply the characterization to computational complexity of identity checking, finite axiomatizability of equational theories, and algebraic descriptions of certain classes of recognizable languages.

Theorems & Definitions (38)

• Example 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:ais-inequality']}
• Corollary 2.3
• Example 2.4
• proof
• Corollary 2.5
• proof