AI 中文总结
研究扩展广义排列多面体上共作用双代数结构，通过测量代数框架找到合适共作用概念，证明EGP和仿射锥EGP形成共作用双幺半群，明确其与扩展次模函数对应关系及面和切锥的次模函数，辫子扇等起关键作用。
AI 中文摘要
最近，扩展广义排列多面体（EGP）上的霍普夫幺半群结构被引入。本文研究EGP上共作用双代数结构的存在性。结果表明，合适的共作用概念存在，并非经典余模意义下的，而是通过测量代数框架。余模型映射为每个多面体赋予面与面处切锥的对之和。EGP和仿射锥EGP在物种中形成共作用双幺半群，EGP与扩展次模函数一一对应，还明确描述了其面和切锥的次模函数，辫子扇及其与预序的关系在此描述中起关键作用。
英文摘要
A Hopf monoid structure on extended generalized permutahedra (EGP) was recently introduced by M.Aguiar and F.Ardila. We investigate the existence of a cointeracting bialgebra structure on EGP's. We show that a suitable notion of cointeraction exists, not in the classical comodule sense, but via the framework of measuring algebras. The comodule-type map assigns to each polyhedron the sum of pairs of face and tangent cone at the face. EGP's and affine cone EGP's form the cointeracting bimonoids in species with EGP as a third measuring structure. EGP's are in bijection to extended submodular functions. For an EGP, we also describe explicitly the submodular functions of its faces and tangent cones. The braid fan and its relation to preorders play a key role in this description.
CommentsThis is a substantial rewriting of an unpublished preprint "Submodular functions, generalized permutahedra, confomring preorders, and cointeracting bialgebras". This new and shortened article shifts focus to polyhedral geometry, and introduces the notion of measuring algebra for the cointeraction of bialgebras
