论文标题
扭曲分区的一致性及其晶格的属性
Properties of congruences of twisted partition monoids and their lattices
论文作者
论文摘要
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.我们证明，$ \ Mathcal {p} _ {n}^φ$的每个一致性（有限）最多是由$ \ lceil \ frac {5n} 2 \ rceil $ Pairs生成的，我们表征了主要的。我们还证明，一致性晶格$ \ textsf {cong}（\ mathcal {p} _ {n}^φ）$不是模块化（或分配）;它没有无限上升的链条，但确实具有无限的下降链和无限的敌人。相比之下，晶格$ \ textsf {cong}（\ Mathcal {p} _ {n，d}^φ）$是模块化的，但仍未分配$ d> 0 $，而$ \ textsf {cong}（\ natercal {\ nathcal {p} _ {n，0} _ {n，0}^φ）$是分配。我们还计算了$ \ MATHCAL {p} _ {n，d}^φ$的一致性，表明数组$ \ big（| \ textsf {cong}（\ Mathcal {p} _ {p} _ {n，d}^φ） $ d $，$ | \ textsf {cong}（\ Mathcal {p} _ {n，d}^φ）| $分别是$ d $或$ n \ geq 4 $中的多项式。
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.
