链的内态单元直径

James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson
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引用次数: 0

摘要

单元的左直径和右直径是拓扑不变量,定义为与通用左或右全同的有限生成集有关的派生序列长度的上值。我们为链$C$的内态单元$End(C)$计算这些参数。具体地说,如果$C$是无限的,那么$End(C)$的左直径是2,而右直径要么是2要么是3,后者恰恰等于2,即当$C$是某个端点$z$的$C{\setminus}\{z\}$的上簇时。如果$C$是有限的,那么$End(C)$也是有限的,在这种情况下,左右直径分别为1(如果$C$不是三维的)或0.
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Diameters of endomorphism monoids of chains
The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{\setminus}\{z\}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0.
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