Properties of congruences of twisted partition monoids and their lattices
Properties of congruences of twisted partition monoids and their lattices
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扭曲分块幺半群及其格的同余性质
DOI：
10.1112/jlms.12574
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
East J
中科院分区：
文献类型：
--
作者：
East J
We build on the recent characterisation of congruences on the infinite twisted partition monoids PnΦ$\mathcal {P}_{n}^\Phi$ and their finite d$d$‐twisted homomorphic images Pn,dΦ$\mathcal {P}_{n,d}^\Phi$, and investigate their algebraic and order‐theoretic properties. We prove that each congruence of PnΦ$\mathcal {P}_{n}^\Phi$ is (finitely) generated by at most ⌈5n2⌉$\lceil \frac{5n}{2}\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong(PnΦ)$\text{\sf Cong}(\mathcal {P}_{n}^\Phi )$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti‐chains. By way of contrast, the lattice Cong(Pn,dΦ)$\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )$ is modular but still not distributive for d>0$d>0$, while Cong(Pn,0Φ)$\text{\sf Cong}(\mathcal {P}_{n,0}^\Phi )$ is distributive. We also calculate the number of congruences of Pn,dΦ$\mathcal {P}_{n,d}^\Phi$, showing that the array (|Cong(Pn,dΦ)|)n,d⩾0$(|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|)_{n,d\geqslant 0}$ has a rational generating function, and that for a fixed n$n$ or d$d$, |Cong(Pn,dΦ)|$|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|$ is a polynomial in d$d$ or n⩾4$n\geqslant 4$, respectively.
We build on the recent characterisation of congruences on the infinite twisted partition monoids PnΦ$\mathcal {P}_{n}^\Phi$ and their finite d$d$‐twisted homomorphic images Pn,dΦ$\mathcal {P}_{n,d}^\Phi$, and investigate their algebraic and order‐theoretic properties. We prove that each congruence of PnΦ$\mathcal {P}_{n}^\Phi$ is (finitely) generated by at most ⌈5n2⌉$\lceil \frac{5n}{2}\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong(PnΦ)$\text{\sf Cong}(\mathcal {P}_{n}^\Phi )$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti‐chains. By way of contrast, the lattice Cong(Pn,dΦ)$\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )$ is modular but still not distributive for d>0$d>0$, while Cong(Pn,0Φ)$\text{\sf Cong}(\mathcal {P}_{n,0}^\Phi )$ is distributive. We also calculate the number of congruences of Pn,dΦ$\mathcal {P}_{n,d}^\Phi$, showing that the array (|Cong(Pn,dΦ)|)n,d⩾0$(|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|)_{n,d\geqslant 0}$ has a rational generating function, and that for a fixed n$n$ or d$d$, |Cong(Pn,dΦ)|$|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|$ is a polynomial in d$d$ or n⩾4$n\geqslant 4$, respectively.
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影响因子：
0.8
作者：
J. East;N. Ruškuc
通讯作者：
J. East;N. Ruškuc
影响因子：
1.7
作者：
J. East;N. Ruškuc
通讯作者：
J. East;N. Ruškuc
影响因子：
1.7
作者：
J. East;J. Mitchell;N. Ruškuc;M. Torpey
通讯作者：
J. East;J. Mitchell;N. Ruškuc;M. Torpey
DOI：
10.1007/978-1-84800-281-4
发表时间：
2008-12
期刊：
--
影响因子：
--
作者：
O. Ganyushkin;V. Mazorchuk
通讯作者：
O. Ganyushkin;V. Mazorchuk
关于分区幺半群和一些相关的半群
影响因子：
0.7
作者：
D. Fitzgerald;K. Lau
通讯作者：
K. Lau