On the lattice of multi-sorted relational clones on a two-element set
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914175386648576 |
|---|---|
| author | David, Vojtěch Zhuk, Dmitriy |
| author_facet | David, Vojtěch Zhuk, Dmitriy |
| contents | We introduce a new approach to the description of multi-sorted clones (sets of $k$-tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the well-known Galois connection between operations and relations, we define a small class of canonical relations sufficient to describe all Boolean multi-sorted clones up to non-surjective operations. Furthermore, we introduce elementary operations on relations, which are less cumbersome than general formulas and have many useful properties. Using these tools, we provide a new and elementary proof of the famous Post's lattice theorem. We also show that every multi-sorted clone of $k$-tuples of operations decomposes into a surjective part described by canonical relations and $2k$ clones of $(k-1)$-tuples of operations. This structural understanding allows us to describe an embedding of the lattice of multi-sorted clones into a well-understood poset. In particular, we rederive - by a simpler method - a result of V. Taimanov originally from 1983, showing that every multi-sorted clone on a two-element domain is finitely generated. Finally, we also give a concise proof of the Galois connection between (surjective) multi-sorted clones and the corresponding closed sets of relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the lattice of multi-sorted relational clones on a two-element set David, Vojtěch Zhuk, Dmitriy Logic Rings and Algebras 08A40, 08A02, 06E30 We introduce a new approach to the description of multi-sorted clones (sets of $k$-tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the well-known Galois connection between operations and relations, we define a small class of canonical relations sufficient to describe all Boolean multi-sorted clones up to non-surjective operations. Furthermore, we introduce elementary operations on relations, which are less cumbersome than general formulas and have many useful properties. Using these tools, we provide a new and elementary proof of the famous Post's lattice theorem. We also show that every multi-sorted clone of $k$-tuples of operations decomposes into a surjective part described by canonical relations and $2k$ clones of $(k-1)$-tuples of operations. This structural understanding allows us to describe an embedding of the lattice of multi-sorted clones into a well-understood poset. In particular, we rederive - by a simpler method - a result of V. Taimanov originally from 1983, showing that every multi-sorted clone on a two-element domain is finitely generated. Finally, we also give a concise proof of the Galois connection between (surjective) multi-sorted clones and the corresponding closed sets of relations. |
| title | On the lattice of multi-sorted relational clones on a two-element set |
| topic | Logic Rings and Algebras 08A40, 08A02, 06E30 |
| url | https://arxiv.org/abs/2505.06033 |