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Abstract
We build on the recent characterisation of congruences on the infinite twisted partition monoids P^{Φ}_{n} and their finite dtwisted homomorphic images P^{Φ}_{n,d}, and investigate their algebraic and ordertheoretic properties. We prove that each congruence of P^{Φ}_{n} is (finitely) generated by at most ⌈5n/2⌉ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong(P^{Φ}_{n}) 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 Cong(P^{Φ}_{n,d}) is modular but still not distributive for d>0, while Cong(P^{Φ}_{n,0}) is distributive. We also calculate the number of congruences of P^{Φ}_{n,d}, showing that the array (Cong(P^{Φ}_{n,d}))_{n,d≥0} has a rational generating function, and that for a fixed n or d, Cong(P^{Φ}_{n,d}) is a polynomial in d or n≥4, respectively.
Original language  English 

Pages (fromto)  311357 
Journal  Journal of the London Mathematical Society 
Volume  106 
Issue number  1 
Early online date  15 Mar 2022 
DOIs  
Publication status  Published  Jul 2022 
Keywords
 Partition monoid
 Twisted partition monoid
 Congruence
 Finitely generated congruence
 Principal congruence
 Congruence lattice
 Modular lattice
 Distributive lattice
 Enumeration
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Dive into the research topics of 'Properties of congruences of twisted partition monoids and their lattices'. Together they form a unique fingerprint.Projects
 1 Finished

Diagram Monoids and Their Congruences: Diagram Monoids and Their Congruences
15/12/18 → 14/02/21
Project: Standard