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- W2802152000 abstract "In the paper we describe the group $mathbf{Aut}left(mathscr{C}_{mathbb{Z}}right)$ of automorphisms of the extended bicyclic semigroup $mathscr{C}_{mathbb{Z}}$ and study the variants $mathscr{C}_{mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $mathscr{C}_{mathbb{Z}}$, where $m,ninmathbb{Z}$. In particular, we prove that $mathbf{Aut}left(mathscr{C}_{mathbb{Z}}right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup $mathscr{C}_{mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E(mathscr{C}_{mathbb{Z}}^{m,n})$ and Green's relations on the semigroup $mathscr{C}_{mathbb{Z}}^{m,n}$. Also we show that $E(mathscr{C}_{mathbb{Z}}^{m,n})$ is an $omega$-chain and any two variants of the extended bicyclic semigroup $mathscr{C}_{mathbb{Z}}$ are isomorphic. At the end we discuss shift-continuous Hausdorff topologies on the variant $mathscr{C}_{mathbb{Z}}^{0,0}$. In particular, we prove that if $tau$ is a Hausdorff shift-continuous topology on $mathscr{C}_{mathbb{Z}}^{0,0}$ then each of inequalities $a>0$ or $b>0$ implies that $(a,b)$ is an isolated point of $big(mathscr{C}_{mathbb{Z}}^{0,0},taubig)$ and construct an example of a Hausdorff semigroup topology $tau^*$ on the semigroup $mathscr{C}_{mathbb{Z}}^{0,0}$ such that all its points with $ableqslant 0$ and $a+bleqslant 0$ are not isolated in $big(mathscr{C}_{mathbb{Z}}^{0,0},tau^*big)$." @default.
- W2802152000 created "2018-05-17" @default.
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- W2802152000 date "2018-05-14" @default.
- W2802152000 modified "2023-09-23" @default.
- W2802152000 title "On variants of the extended bicyclic semigroup" @default.
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