The Thins Ordering on Relations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914851608068096 |
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| author | Voermans, Ed Desharnais, Jules Backhouse, Roland |
| author_facet | Voermans, Ed Desharnais, Jules Backhouse, Roland |
| contents | Earlier papers \cite{VB2022,VB2023a,VB2023b} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial equivalence relations (pers) have an index -- and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the \textsf{thins} ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the \textsf{thins} ordering. We also characterise the relations that are maximal with respect to the \textsf{thins} ordering. Apart from our axiom of choice, the axiom system we employ is paired to a bare minimum and admits many models other than concrete relations -- we do not assume, for example, the existence of complements; in the case of concrete relations, the theorem is that the maximal elements of the \textsf{thins} ordering are the empty relation and the equivalence relations. This and other properties of \textsf{thins} provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Thins Ordering on Relations Voermans, Ed Desharnais, Jules Backhouse, Roland Logic in Computer Science F.3.1; F.4.1 Earlier papers \cite{VB2022,VB2023a,VB2023b} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial equivalence relations (pers) have an index -- and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the \textsf{thins} ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the \textsf{thins} ordering. We also characterise the relations that are maximal with respect to the \textsf{thins} ordering. Apart from our axiom of choice, the axiom system we employ is paired to a bare minimum and admits many models other than concrete relations -- we do not assume, for example, the existence of complements; in the case of concrete relations, the theorem is that the maximal elements of the \textsf{thins} ordering are the empty relation and the equivalence relations. This and other properties of \textsf{thins} provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations. |
| title | The Thins Ordering on Relations |
| topic | Logic in Computer Science F.3.1; F.4.1 |
| url | https://arxiv.org/abs/2401.16888 |