Congruences on bicyclic extensions of a linearly ordered group

On semitopological bicyclic extensions of linearly ordered groups

For a linearly ordered group $G$ let us define a subset $A\subseteq G$ to be a \emph{shift-set} if for any $x,y,z\in A$ with $y < x$ we get $x\cdot y^{-1}\cdot z\in A$. We describe the natural partial order and solutions of equations on the semigroup $\mathscr{B}(A)$ of shifts of positive cones of $A$. We study topologizations of the semigroup $\mathscr{B}(A)$. In particular, we show that for an arbitrary countable linearly ordered group $G$ and a non-empty shift-set $A$ of $G$ every Baire shift-continuous $T_1$-topology $τ$ on $\mathscr{B}(A)$ is discrete. Also we prove that for an arbitrary …

关于阿基米德全序群的双循环扩张的同态

On homomorphisms of bicyclic extensions of archimedean totally ordered groups

本文研究阿基米德全序群双循环扩张上的同态，证明群 $o$-同态与扩张幺半群同态之间可相互生成。

On the closure of the extended bicyclic semigroup

In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\oper…

Perfect congruences on bisimple $ω$-semigroups

A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $ω$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $ω$-semigroups, substantially generalizing the above mentioned …

关于半群 $\mathscr{C}_{+}(a,b)$ 的幺半群自同态半群

On the semigroup of monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$

本文研究双循环幺半群子幺半群 $\mathscr{C}_{+}(a,b)$，刻画了由其全部同余和单射幺半群自同态共同生成的幺半群自同态。

On monoids of injective partial cofinite selfmaps

We study the semigroup $\mathscr{I}^{\mathrm{cf}}_λ$ of injective partial cofinite selfmaps of an infinite cardinal $λ$. We show that $\mathscr{I}^{\mathrm{cf}}_λ$ is a bisimple inverse semigroup and each chain of idempotents in $\mathscr{I}^{\mathrm{cf}}_λ$ is contained in a bicyclic subsemigroup of $\mathscr{I}^{\mathrm{cf}}_λ$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_λ$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_λ$ is a group congruence. Also, we describe the structure of the quotient semigroup $\mathscr{I}^{\mathrm{cf}}_λ/σ$, where $σ…