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Search: MSC category 20M30 ( Representation of semigroups; actions of semigroups on sets )

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1. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords:super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

2. CMB 2011 (vol 55 pp. 188)

Steinberg, Benjamin
Yet Another Solution to the Burnside Problem for Matrix Semigroups
We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Keywords:Burnside problem, kernel category
Categories:20M30, 20M32

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