The category of affine algebraic regular monoids

<abstract><p>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 $ \mathcal{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.</p></abstract>