扭分拆幺半群的同余及其格的性质
Properties of congruences of twisted partition monoids and their lattices
摘要：
我们建立在最近的刻画上的无穷扭曲分割幺半群$\mathcal{P}_{n}^Φ$和他们的有限$d$扭曲同态图像$\mathcal{P}_{n,d}^Φ$的同余，并调查他们的代数和序理论的性质。我们证明了$\mathcal{P}_{n}^Φ$的每一个同余至多由$\lceil\frac{5n}2\rceil$对生成，并证明了主同余.我们还证明了同余格$\textsf{Cong}(\mathcal{P}_{n}^Φ)$不是模的（或分配的），它没有无限上升链，但它有无限下降链和无限反链。作为对比，格$\textsf{Cong}(\mathcal{P}_{n,d}^Φ)$是模块化的，但对于$d>0$仍然不是分配的，而$\textsf{Cong}(\mathcal{P}_{n,0}^Φ)$是分配的。我们还计算了$\mathcal{P}_{n,d}^Φ$的同余数，表明阵列$\big(|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|\big)_{n,d\geq 0}$具有有理生成函数，并且对于固定的$n$或$d$，$|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|$分别是$d$或$n\geq 4$中的多项式。
英文摘要：
We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^Φ$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^Φ$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^Φ$ is (finitely) generated by at most $\lceil\frac{5n}2\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\textsf{Cong}(\mathcal{P}_{n}^Φ)$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite antichains. By way of contrast, the lattice $\textsf{Cong}(\mathcal{P}_{n,d}^Φ)$ is modular but still not distributive for $d>0$, while $\textsf{Cong}(\mathcal{P}_{n,0}^Φ)$ is distributive. We also calculate the number of congruences of $\mathcal{P}_{n,d}^Φ$, showing that the array $\big(|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|\big)_{n,d\geq 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|$ is a polynomial in $d$ or $n\geq 4$, respectively.
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