Large Sets of Lattices without Order Embeddings
Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ℵ0, μ can be constructed in countably many steps by passing from a cardinal λ to 2λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {Li: i ∈ I}...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2016
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| Sorozat: | COMMUNICATIONS IN ALGEBRA
44 No. 2 |
| doi: | 10.1080/00927872.2014.967352 |
| mtmt: | 3014329 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/14544 |
| Tartalmi kivonat: | Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ℵ0, μ can be constructed in countably many steps by passing from a cardinal λ to 2λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {Li: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: Li → Lj is an order embedding, then i = j and φ is the identity map of Li. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices. © 2016, Copyright © Taylor & Francis Group, LLC. |
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| Terjedelem/Fizikai jellemzők: | 668-679 |
| ISSN: | 0092-7872 |