Lattice Homomorphisms Between Weak Orders

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.

Weak orders on symmetric groups and posets of support τ-tilting modules

This paper treats algebras whose support [Formula: see text]-tilting posets are isomorphic to the poset of the symmetric group with the weak order. We determine such algebras in terms of quivers and relations.

Representability of Fuzzy Biorders and Fuzzy Weak Orders

In this paper, we have studied representability of both fuzzy biorders and fuzzy weak orders. It has also been shown that union of a finite family of fuzzy weak orders with respect to a t-norm T is a fuzzy quasi-transitive relation with respect to T. In the last theorem, we have obtained a characterization for a TL-representable fuzzy weak order.