On Pleasant Eulerian posets Subbarayan, R.; Vethamanickam, A.; Amirtha, V. Gladys Mano
Discussiones mathematicae. General algebra and applications,
2023, Letnik:
43, Številka:
1
Journal Article
We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set ...of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.
As a common generalization of s2-quasicontinuous posets and quasi Z-continuous domains, the concept of sZ-quasicontinuous posets is introduced and some of their basic properties are investigated. It ...is proved that if a subset system Z satisfies certain conditions, and P is an sZ-quasicontinuous poset, then the Z-way below relation ≪Z on P has the interpolation property, the space (P,σZ(P)) is locally compact and the space (P,λZ(P)) is a pospace. It is also proved that under some conditions, a poset is sZ-continuous if and only if it is meet sZ-continuous and sZ-quasicontinuous.
Ordered weighted averaging (OWA) operators, a family of aggregation functions, are widely used in human decision-making schemes to aggregate data inputs of a decision maker's choosing through a ...process known as OWA aggregation. The weight allocation mechanism of OWA aggregation employs the principle of linear ordering to order data inputs after the input variables have been rearranged. Thus, OWA operators generally cannot be used to aggregate a collection of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> inputs obtained from any given convex partially ordered set (poset). This poses a problem since data inputs are often obtained from various convex posets in the real world. To address this problem, this paper proposes methods that practitioners can use in real-world applications to aggregate a collection of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> inputs from any given convex poset. The paper also analyzes properties related to the proposed methods, such as monotonicity and weighted OWA aggregation on convex posets.
We prove a version of the fundamental theorems of Morse theory in the setting of finite partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell ...complexes and derive the Morse-Pitcher inequalities in that context.
We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this ...relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave "as if" they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelgänger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelgänger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type).
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