Algebra in superextensions of semilattices
Given a semilattice \(X\) we study the algebraic properties of the semigroup \(\upsilon(X)\) of upfamilies on \(X\). The semigroup \(\upsilon(X)\) contains the Stone-Cech extension \(\beta(X)\), the superextension \(\lambda(X)\), and the space of filters \(\varphi(X)\) on \(X\) as closed subsemigrou...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/690 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Given a semilattice \(X\) we study the algebraic properties of the semigroup \(\upsilon(X)\) of upfamilies on \(X\). The semigroup \(\upsilon(X)\) contains the Stone-Cech extension \(\beta(X)\), the superextension \(\lambda(X)\), and the space of filters \(\varphi(X)\) on \(X\) as closed subsemigroups. We prove that \(\upsilon(X)\) is a semilattice iff \(\lambda(X)\) is a semilattice iff \(\varphi(X)\) is a semilattice iff the semilattice \(X\) is finite and linearly ordered. We prove that the semigroup \(\beta(X)\) is a band if and only if \(X\) has no infinite antichains, and the semigroup \(\lambda(X)\) is commutative if and only if \(X\) is a bush with finite branches. |
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