Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.08841 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- We investigate the Mal'tsev product $\mathcal{V} \circ \mathcal{W}$ of two varieties $\mathcal{V}$ and $\mathcal{W}$ of the same similarity type. Such a product is usually a quasivariety but not necessarily a variety. We give an equational base for the variety generated by $\mathcal{V} \circ \mathcal{W}$ in terms of identities satisfied in $\mathcal{V}$ and $\mathcal{W}$. Then the main result provides a new sufficient condition for $\mathcal{V} \circ \mathcal{W}$ to be a variety: If $\mathcal{W}$ is an idempotent variety and there are terms $f(x,y)$ and $g(x,y)$ such that $\mathcal{W}$ satisfies the identity $f(x,y) = g(x,y)$ and $\mathcal{V}$ satisfies the identities $f(x,y) = x$ and $g(x,y) = y$, then $\mathcal{V} \circ \mathcal{W}$ is a variety. We also provide a number of examples and applications of this result.