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This article is cited in 1 scientific paper (total in 1 paper)
RESEARCH ARTICLE
On the lattice of weak topologies on the bicyclic monoid with adjoined zero
S. Bardylaa, O. Gutikb a Institute of Mathematics, Kurt Gödel Research Center, Vienna, Austria
b Department of Mechanics and Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine
Abstract:
A Hausdorff topology $\tau$ on the bicyclic monoid with adjoined zero $\mathcal{C}^0$ is called weak if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice $\mathcal{SIF}^1\times\mathcal{SIF}^1$ where $\mathcal{SIF}^1$ is the set of all shift-invariant filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ if and only if $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we prove that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.
Keywords:
lattice of topologies, bicyclic monoid, shift-continuous topology.
Received: 17.09.2019 Revised: 26.11.2019
Citation:
S. Bardyla, O. Gutik, “On the lattice of weak topologies on the bicyclic monoid with adjoined zero”, Algebra Discrete Math., 30:1 (2020), 26–43
Linking options:
https://www.mathnet.ru/eng/adm763 https://www.mathnet.ru/eng/adm/v30/i1/p26
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