# Directional Non-Commutative Monoidal Structures for Compositional Embeddings in Machine Learning

arXiv: https://arxiv.org/abs/2505.15507 · HTML: https://mlsanity.com/paper/2505.15507/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 21 May 2025.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)

## Built on by (most cited first, 4 of 4)

- [Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators](https://mlsanity.com/paper/2505.24533/) (2025)
- [JoFormer (Journey-based Transformer): Theory and Empirical Analysis on the Tiny Shakespeare Dataset](https://mlsanity.com/paper/2506.08652/) (2025)
- [Directional Non-Commutative Monoidal Embeddings for MNIST](https://mlsanity.com/paper/2506.03472/) (2025)
- [Commutator-Induced Uncertainty in VAEs](https://mlsanity.com/paper/2605.23449/) (2026)

## Abstract (arXiv)

We introduce a new algebraic structure for multi-dimensional compositional embeddings, built on directional non-commutative monoidal operators. The core contribution of this work is this novel framework, which exhibits appealing theoretical properties (associativity along each dimension and an interchange law ensuring global consistency) while remaining compatible with modern machine learning architectures. Our construction defines a distinct composition operator circ_i for each axis i, ensuring associative combination along each axis without imposing global commutativity. Importantly, all axis-specific operators commute with one another, enforcing a global interchange law that enables consistent crossaxis compositions. This is, to our knowledge, the first approach that provides a common foundation that generalizes classical sequence-modeling paradigms (e.g., structured state-space models (SSMs) and transformer self-attention) to a unified multi-dimensional framework. For example, specific one-dimensional instances of our framework can recover the familiar affine transformation algebra, vanilla self-attention, and the SSM-style recurrence. The higher-dimensional generalizations naturally support recursive, structure-aware operations in embedding spaces. We outline several potential applications unlocked by this structure-including structured positional encodings in Transformers, directional image embeddings, and symbolic modeling of sequences or grids-indicating that it could inform future deep learning model designs. We formally establish the algebraic properties of our framework and discuss efficient implementations. Finally, as our focus is theoretical, we include no experiments here and defer empirical validation to future work, which we plan to undertake.