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An infinite chain and an infinite anti-chain of compactness principles related to directed cycles and finite choice are found related to directed cycles and finite choice.
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\mathcal{C}_N\mathfrak{g}=\mathfrak{g}\otimes (\mathbb{C}[t]/t^{N+1}\mathbb{C}[t])$ denote the $N$-th Takiff algebra of $\mathfrak{g}$. We introduce a notion of parabolic subalgebras of $\mathcal{C}_N\mathfrak{g}$ and classify them in t...
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