COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS

Proceedings of the Edinburgh Mathematical Society

P. M. Higgins¹ 

[1] J. M. Howie

关键词: transformation semigroups;    rank;    countable;    binary relations;   

DOI  :  10.1017/S0013091502000974

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

The relative rank $ank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $langle Acup Bangle=S$. It follows from a result of Sierpiński that, if $X$ is infinite, the relative rank of a subset of the full transformation semigroup $mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $mathcal{B}_{X}$ of binary relations on $X$.A subset $S$ of $mathcal{T}_{mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $mathcal{T}_{mathbb{N}}$ with the property that for each $sigma$ in $S$ there exists $mu$ in $U$ such that $isigmale imu$ for all $i$ in $mathbb{N}$. It is shown that every dominated subset of $mathcal{T}_{mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid of all contractions in $mathcal{T}_{mathbb{N}}$ (mappings $alpha$ with the property that $|ialpha-jalpha|le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.It is shown (among other results) that $ank(mathcal{B}_{X}:mathcal{T}_{X})=1$ and that $ank(mathcal{B}_{X}:mathcal{I}_{X})=1$ (where $mathcal{I}_{X}$ is the symmetric inverse semigroup on $X$). By contrast, if $mathcal{S}_{X}$ is the symmetric group, $ank(mathcal{B}_{X}:mathcal{S}_{X})=2$.AMS 2000 Mathematics subject classification: Primary 20M20

【 授权许可】

Unknown