Finitary Cartesian closed varieties and semigroup actions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910030230454272 |
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| author | Lawson, Mark V |
| author_facet | Lawson, Mark V |
| contents | We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19904 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finitary Cartesian closed varieties and semigroup actions Lawson, Mark V Category Theory We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid. |
| title | Finitary Cartesian closed varieties and semigroup actions |
| topic | Category Theory |
| url | https://arxiv.org/abs/2602.19904 |