@article{Cao2022, 
author = {Haijun Cao and Fang Xiao},
title = {The category of affine algebraic regular monoids},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {2},
pages = {2666-2679},
keywords = {weak Hopf algebra, affine algebraic regular monoid, affine weak Hopf algebra},
url = {https://www.sciopen.com/article/10.3934/math.2022150},
doi = {10.3934/math.2022150},
abstract = {The main aim of this study is to characterize affine weak    k-algebra    H whose affine    k-variety    S  =      M          k        (  H  ,  k  ) admits a regular monoid structure. As preparation, we determine some results of weak Hopf algebras morphisms, and prove that the anti-function from the category        C   of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint. Then, we prove the main result of this study: the anti-equivalence between the category of affine algebraic    k-regular monoids and the category of finitely generated commutative reduced weak    k-Hopf algebras.}
}