If \({\mathbf {G}}=(G;\cdot )\) is a semigroup, I is arbitrary set and \(\lambda ,\rho :I\rightarrow I\) are mappings satisfying the equalities \(\lambda \lambda =\lambda \), \(\rho \rho =\rho \) and \(\lambda \rho =\rho \lambda \) then we define the semigroup \((G^I,\times )\) where \((x\times y)(i) := x( \lambda i)\cdot y (\rho i)\). This construction gives rise to four covariant and two contravariant functors and constitute three adjoint situations. We apply this functors for finding representation theorems.

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关于由集合上的交换回缩引起的半群构造

如果\({\mathbf {G}}=(G;\cdot )\)是一个半群，I是任意集，而\(\lambda ,\rho :I\rightarrow I\)是满足等式\(\ lambda \lambda =\lambda \) , \(\rho \rho =\rho \)和\(\lambda \rho =\rho \lambda \)然后我们定义半群\((G^I,\times )\ )其中\((x\times y)(i) := x( \lambda i)\cdot y (\rho i)\)。这种构造产生四个协变和两个逆变函子并构成三个伴随情况。我们应用这个函子来寻找表示定理。