# Hereditarily antisymmetric operator algebras

## TL;DR

This paper introduces and characterizes hereditarily antisymmetric operator algebras, providing structure theorems in finite dimensions and partial results for infinite dimensions, with implications for matrix representations and poset theorems.

## Contribution

It defines hereditarily antisymmetric operator algebras and proves a structure theorem for finite dimensions, extending classical poset theorems to matrix analogs.

## Key findings

- Finite-dimensional hereditarily antisymmetric operator algebras have a specific structure.
- Characterization of operator algebras that can be made upper triangular.
- Matrix analogs of Dilworth and Mirsky theorems for finite posets.

## Abstract

We introduce a notion of ``hereditarily antisymmetric'' operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.