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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.08841 |
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| _version_ | 1866913314151333888 |
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| author | Penza, Tomasz Romanowska, Anna B. |
| author_facet | Penza, Tomasz Romanowska, Anna B. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mal'tsev products of varieties, I Penza, Tomasz Romanowska, Anna B. Rings and Algebras 08B05 (Primary) 08C15, 08A30 (Secondary) 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. |
| title | Mal'tsev products of varieties, I |
| topic | Rings and Algebras 08B05 (Primary) 08C15, 08A30 (Secondary) |
| url | https://arxiv.org/abs/2404.08841 |