论文标题
零关联的半群的一层
Polyboundedness of zero-closed semigroups
论文作者
论文摘要
半群$ x $的多基数$ \ mathrm {cov}（\ mathcal a_x）$是$ x $的最小基数，$ x $ by x $ $ \ {x \ in x：a_0xa_1 $ a_0，\ dots，a_n \ in x^1 = x \ cup \ {1 \} $。具有有限多层数字的半群称为一层。如果$ x $在其$ 0 $ -EXTENSION $ x^0 = \ {0 \} \ cup x $中，则为零关闭的Semigroup $ x $被称为零关闭。我们证明，任何零关闭的无限semogroup $ x $都有$ \ mathrm {cov}（\ mathcal a_x）<| x | $。在马丁的公理下，如果$ x $承认紧凑的hausdorff semigroup拓扑或$ x $具有可分离的完整次变体度量，则将零关闭的半群套在一起。
The polyboundedness number $\mathrm{cov}(\mathcal A_X)$ of a semigroup $X$ is the smallest cardinality of a cover of $X$ by sets of the form $\{x\in X:a_0xa_1\cdots xa_n=b\}$ for some $n\ge 1$, $b\in X$ and $a_0,\dots,a_n\in X^1=X\cup\{1\}$. Semigroups with finite polyboundedness number are called polybounded. A semigroup $X$ is called zero-closed if $X$ is closed in its $0$-extension $X^0=\{0\}\cup X$ endowed with any Hausdorff semigroup topology. We prove that any zero-closed infinite semigroup $X$ has $\mathrm{cov}(\mathcal A_X)<|X|$. Under Martin's Axiom, a zero-closed semigroup is polybounded if $X$ admits a compact Hausdorff semigroup topology or $X$ has a separable complete subinvariant metric.
