Structure of the rational monoid algebra for Boolean matrices of order 3

TL;DR

This paper uses computer algebra to analyze the structure of the rational monoid algebra of 3x3 Boolean matrices, revealing its radical, center, primitive idempotents, and irreducible representations.

Contribution

It provides a detailed structural analysis of the 3x3 Boolean matrix monoid algebra, including basis, radical, center, primitive idempotents, and irreducible representations, using computational methods.

Findings

Basis for the radical linked to non-regular elements

Center of the semisimple quotient has dimension 14

Semisimple quotient decomposes into matrix algebras of specified sizes

Abstract

We use computer algebra to study the 512-dimensional associative algebra Q B_3, the rational monoid algebra of 3 x 3 Boolean matrices. We obtain a basis for the radical in bijection with the 42 non-regular elements of B_3. The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to d x d rational matrix algebras for d = 1, 1, 1, 2, 3, 3, 3, 3, 6, 6, 7, 9, 9, 12. We construct the irreducible representations of B_3 over Q by calculating the representation matrices for a minimal set of generators.