The category of affine algebraic regular monoids

1.   Introduction

The definition of a monoidal category first appeared in 1963 in the work of MacLane ^[8], and later in his classic book ^[9] (first published in 1971). The notion of a Hopf algebra first appeared in topology (more specifically, in the work ^[5] of Hopf in 1941, as an algebraic construction capturing the structure of cohomology rings of H-spaces, in particular, Lie groups). Around that time, it was realized that Hopf algebras could be viewed as algebraic structures arising from tensor categories with a fiber functor, i.e., a tensor functor to the category of vector spaces, through the so-called reconstruction theory (which takes its origins in ^[4,19]). Since then, the theory of Hopf algebras has gradually became a part of the theory of tensor categories, and the theory of categories has been used in proving some of the more recent results on Hopf algebras (such as the classification of semisimple Hopf algebras of prime power dimension and the classification of triangle Hopf algebras). Linear algebraic groups have been studied by a number of well known mathematicians ^[2,6] for details. Very special classes of linear algebraic semigroups were studied in detail in ^[3,14,15]. A general study of linear algebraic semigroups was begun by Mohan. S. Putcha in the papers ^[16,17]. As known, the category of affine algebraic groups is anti-equivalent to the category of finitely generated commutative reduced Hopf algebras ^[1]. The weak Hopf algebra proposed by Fang Li is an extension of Hopf algebras. So, we can define the so-called affine algebraic regular monoid such that the category of affine algebraic regular monoids is anti-equivalent to the category of finitely generated commutative reduced weak Hopf algebras. This is the main work of this paper.

2.   Preliminary

Throughout this paper, $\bf{R}, \bf{Z^{+}}$ will denote the sets of all reals and positive integers, respectively.

As preparation, we introduce several concepts that related to a semigroup $S.$ For an element $x\in S,$ we call $x$ regular if there exists some $x^{*}\in S$ satisfying

$x^{*}$ is called the regular inverse of $x.$ We denote

the set of all regular inverse elements of $x$ and

the set of all idempotents of $S.$ If all the elements of $S$ are regular, we call $S$ a regular semigroup. If $V(S)$ has only one element for any $x\in S,$ $S$ is called an inverse semigroup; in this instance, we usually denote $x^{-1} = x^{*}.$ A bialgebra $H$ over $k$ is called a weak Hopf algebra ^[10] if there exists $T\in {\rm Hom}_{k}(H, H)$ (the convolution algebra) satisfying

where $T$ is called a weak antipode of $H.$

The same name weak Hopf algebra was also used as another kind of generalization of Hopf algebra in ^[7,12,13] which comultiplication is no longer required to preserve the unit equivalently, the counit is not required to be an algebra homomorphism. We must point out that these two kinds of weak Hopf algebra are completely different generalizations of each other in various directions, since the only common subclass just consists of Hopf algebras see ^[11]. The initial motivation of the latter was its connection with the theory of algebra extension.

$k$ will denote a fixed algebraically closed field, $k^{*} = k\setminus \{0\}$. If $n\in \bf{Z^{+}}$, then $k^{n} = k\times k\times\cdots \times k$ is the affine n-space, $M_{n}(k)$ the multiplicative monoid of all $n\times n$ matrices over $k$ and $GL(n, k)$ its group units. If $X\subset k^{n}$, the $X$ is closed if it is the set of zeroes of a system of polynomials on $k^{n}$. If $X\subset k^{n}$, the $\bar{X}$ denotes the closure of $X$. By an algebraic semigroup we mean a closed subset of $k^{n}$ along with an associative operation which is also a polynomial map. By a homomorphism between algebraic semigroups, we mean a semigroup homomorphism which is also a morphism of varieties (a polynomial map). By a connected semigroup we mean an algebraic semigroup such that the underlying closed set is irreducible (i.e. is not a union of two proper closed subsets).

3.   Affine algebraic semigroup

Firstly, comparing with the concept of affine algebraic group, we will introduce the concept of affine algebraic semigroup.

Definition 2.1. ^[1] If $A$ is a commutative $k$-algebra, the set $X_{A} = M_{k}(A, k)$ of all $k$-algebra morphisms from $A$ to $k$ is called an affine $k$-variety.

In particular, when $A$ is a finitely generated $k$-algebra, we say that $X_{A}$ is an affine algebraic $k$-variety.

Definition 2.2. ^[1] Let $A, B$ be commutative $k$-algebras, and let $\mu:A \rightarrow B$ be a $k$-algebra morphism. Then the map

is called a morphism of affine $k$-varieties.

Definition 2.3. ^[1] If $A$ is a commutative $k$-algebra,

If nil$A = \{0\}$, $A$ is called reduced.

Let $A, B$ be commutative $k$-algebras, $V_{k}$ denotes the category of affine $k$-varieties, and the set of all morphisms from an affine $k$-variety $X_{B}$ to $X_{A}$ is written by $V_{k}(X_{B}, X_{A})$. So, we can get the following result.

Proposition 2.4. ^[1] Let $A$ be a commutative $k$-algebra, the morphism from the affine $k$-variety $X_{A}$ to the affine line $k$ is called a function on $X_{A}$, the set of all functions on $X_{A}$ is written by $V_{k}(X_{A}, k)$. When $A$ is reduced, then $V_{k}(X_{k}, A)\cong A$. The two correspondences defined by

which give an anti-equivalence between the category of finitely generated commutative reduced $k$-algebras and the category of affine algebraic $k$-varieties.

Theorem 2.1. ^[1] If $A, B$ are commutative $k$-algebras, we get

Therefore, the product $X_{A}\times X_{B}$ of $X_{A}$ and $X_{B}$ is taken as sets which admits a structure of an affine $k$-variety, and is called the product of $X_{A}$ and $X_{B}$. If $A$ and $B$ are affine $k$-algebras, $A\otimes_{k} B$ is also an affine $k$-algebra.

Definition 2.5. ^[1] A finitely generated commutative reduced $k$-algebra is called an affine $k$-algebra.

Definition 2.6. ^[1] Given an affine $k$-variety $G = M_{k}(H, k)$ with $H$ is an affine $k$-algebra, $G$ admits a group structure, if the two maps

are morphisms of affine $k$-varieties, then we call $G$ an affine $k$-group.

In particular, when $G$ is an affine algebraic $k$-variety, $G$ is called an affine algebraic $k$-group. The map $m, s$ and the embedding $e\rightarrow G$ of the identity elements $e$ into $G$, correspond to the $k$-algebra morphisms

In this situation, to say that the multiplication map $m: G\times G\rightarrow G$ satisfies the group axioms is equivalent to say that $\Delta$ and $\varepsilon$ satisfy the axioms of $k$-coalgebras and that $S$ is the antipode, in which case, $H$ turns out to be a $k$-Hopf algebra. Therefore an affine $k$-group is precisely $G = M_{k}(H, k)$, where $H$ is commutative reduced $k$-Hopf algebra, and $G$ is regarded as a group where the multiplication is given via the convolution defined by

In other words, $G$ is the group $G(H^{\circ})$ of all group-like elements of the dual k-Hopf algebra $H^{\circ}$ of $H$, where

Then the two correspondences

give an anti-equivalence between the category of affine algebraic $k$-groups and the category of finitely generated commutative reduced $k$-Hopf algebras.

Definition 2.7. ^[1] A $k$-Hopf algebra whose underlying $k$-algebra is an affine $k$-algebra is called an affine $k$-Hopf algebra.

Definition 2.8. A weak $k$-Hopf algebra whose underlying $k$-algebra is an affine $k$-algebra is called an affine weak $k$-Hopf algebra.

It is known that for every regular monoid $S$, its semigroup algebra $kS$ over $k$ is a weak Hopf algebra as generalization of a group algebra. On the other hand, the set of all group-like elements of a weak Hopf algebra is a regular monoid. So, we can give the definition of affine algebra $k$-regular monoid.

Definition 2.9. Given an affine $k$-algebra $H$, the affine $k$-variety $S = M_{k}(H, k)$ admits a regular monoid structure, if the two maps

are morphisms of affine $k$-varieties, then we call $S$ an affine $k$-regular monoid.

Then, as the result of $k$-Hopf algebra $H$ and affine $k$-group $G = M_{k}(H, k)$, we can ge the main result of this paper:

Theorem 2.2. (Main result) Let $k$ be an algebraically closed field, $H$ be a commutative weak $k$-Hopf algebra, the two correspondences

give an anti-equivalence between the category of affine algebraic $k$-regular monoids and the category of finitely generated commutative reduced weak $k$-Hopf algebras.

4.   The proof of main result

Theorem 3.1. ^[10] Let $H = (H, m, \mu, \Delta, \varepsilon, T)$ be a weak $k$-Hopf algebra, for any $a, b\in H$, $T(ab) = T(b)T(a)$, then $H^{\circ} = (H^{\circ}, \Delta^{*}, \varepsilon^{*}, m^{*}, \mu^{*}, T^{*})$ is also a weak $k$-Hopf algebra, where $T^{*}$ is weak Hopf antipode of $H^{\circ}$.

Proposition 3.1. Let $H$ be a weak $k$-Hopf algebra, $A$ be a commutative $k$-algebra, then the set $Alg(H, A)$ of algebra morphisms from $H$ to $A$ is a regular sub-monoid of $Hom(H, A)$.

Proof. Obviously, $\mu_{A}\varepsilon_{H}$ is an algebraic morphism form $H$ to $A$, and it is the identity of $Alg(H, A)$.

Let $f, g\in Alg(H, A)$, $T$ be the weak antipode, for any $x, y\in H$,

So, $f*f\circ T*f = f$, similarly $f\circ T* f*f\circ T = f\circ T$. Hence, $Alg(H, A)$ is a regular sub-monoid of $Hom(H, A)$.

Definition 3.2. ^[10] Let $H$ be a weak Hopf algebra, $I$ is an ideal of $H$.

(1) $\Delta (I)\subseteq H\otimes I + I\otimes H$;

(2) $\varepsilon(I) = 0$;

(3) $T(I)\subseteq I$.

Then, $I$ is called a biideal of $H$ if it satisfies (1) and (2). $I$ is called a weak Hopf ideal of $H$ if it satisfies (1)–(3).

Beishang Ren ^[18] had discussed some properties of quotient Hopf algebra and the Hopf algebra homomorphism by the basic methods of Hopf algebras. Similarly, the following theorem from Theorem 3.2 to Theorem 3.4 will discuss the relative properties about quotient weak Hopf algebra and weak Hopf algebra homomorphism.

Theorem 3.2. Let $K, H, N$ be weak Hopf algebras, $f:K\rightarrow H$, $g:K\rightarrow N$ are weak Hopf algebra morphisms. If $g$ is epimorphism, and $ker(g)\subseteq ker(f)$, then there exists a unique weak Hopf algebra morphism $h:N\rightarrow H$ such that $hg = f$.

Proof. At first, there is a unique $k$-linear map $h:N\rightarrow H$, such that $hg = f$. Since $g$ is epimorphism, for any $n_{1}, n_{2}\in N$, there exist $k_{1}, k_{2}\in K$ such that $n_{1} = g(k_{1}), n_{2} = g(k_{2})$, hence

So, $h$ is an algebra morphism.

On the other hand,

Since $g$ is epimorphism, there is $\Delta_{H}h = (h\otimes h)\Delta_{N}$.

Moreover, $\varepsilon_{H}(hg) = \varepsilon_{H}f = \varepsilon_{K} = \varepsilon_{N}g$, so $\varepsilon_{H}h = \varepsilon_{N}$. Hence, $h$ is a coalgebra morphism.

At last,

that is $T_{H}h = hT_{N}$.

Hence, $h$ is a weak Hopf algebra morphism, and it is unique by the uniqueness of the linear map.

Theorem 3.3. Let $K, H, N$ be weak Hopf algebras, $f:K\rightarrow H$, $h:N\rightarrow H$ are weak Hopf algebra morphisms. If $h$ is monomorphism, and $Im(f)\subseteq Im(h)$, then there exists a unique weak Hopf algebra morphism $g:K\rightarrow N$ such that $hg = f$.

Proof. At first, there is a unique $k$-linear map $g:K\rightarrow N$, such that $hg = f$. Since $f, h$ are algebra morphisms, so, for any $k_{1}, k_{2}\in K$,

Since $h$ is monomorphism, $g(k_{1}k_{2}) = g(k_{1})g(k_{2})$, so h is an algebra morphism.

On the other hand,

Since $h$ is monomorphism, there is $\Delta_{N}g = (g\otimes g)\Delta_{K}$.

Moreover, $\varepsilon_{N}g = \varepsilon_{H}hg = \varepsilon_{H}f = \varepsilon_{K}$, so $\varepsilon_{N}g = \varepsilon_{K}$. Hence, $g$ is a coalgebra morphism.

At last,

that is $gT_{K} = T_{N}g$.

Hence, $g$ is a weak Hopf algebra morphism, and it is unique by the uniqueness of the linear map.

Theorem 3.4. Let $H, L$ be weak Hopf algebras, $\varphi:H\rightarrow L$ is weak Hopf algebra morphism, $I$ is a weak Hopf ideal of $H$ and $\pi: H\rightarrow H/I$ is a regular morphism, then

(1) $Ker\varphi = \{x\in H: \varphi(x) = 0\}$ is a weak Hopf ideal of $H$.

(2) $H/I$ has unique weak Hopf algebra structure such that $\pi$ is a weak Hopf algebra morphism.

(3) For any weak Hopf ideal $I$ which satisfies $I\subseteq Ker\varphi$, there is unique weak Hopf algebra morphism $\bar{\varphi}:H/I\rightarrow L$ such that $\bar{\varphi}\circ \pi = \varphi$.

Proof. (1) $Ker\varphi = \{x\in H: \varphi(x) = 0\}$. For any $x\in Ker\varphi$, $\varphi(x) = 0$, then $(\varphi\otimes \varphi)\Delta_{H}(x) = \Delta_{L}\varphi(x) = 0$. So,

hence, $\Delta_{H}(Ker\varphi)\subseteq Ker\varphi\otimes H+H\otimes Ker\varphi$.

On the other hand, $\varepsilon_{H}(x) = \varepsilon_{L}\varphi(x) = 0$, $\varphi T_{H}(x) = T_{L}\varphi(x) = 0$, so, $T_{H}(x)\in Ker\varphi$, that is $T_{H}(Ker\varphi)\subseteq Ker\varphi$. In conclusion, $Ker\varphi$ is a weak Hopf ideal of $H$.

(2) We can suppose $H = (H, m, \mu, \Delta, \varepsilon, T)$, let $H/I = (H/I, m', \mu', \Delta', \varepsilon', T')$. For any $x, y\in H$, we define

So, it can prove that $H/I = (H/I, m', \mu', \Delta', \varepsilon', T')$ has a weak Hopf algebra structure such that $\pi$ is a weak Hopf algebra morphism.

Suppose $H/I$ has another weak Hopf algebra structure $H/I = (H/I, m_{1}, \mu_{1}, \Delta_{1}, \varepsilon_{1}, T_{1})$ such that $\pi$ is a weak Hopf algebra morphism. Then

Hence, $m' = m_{1}, \mu' = \mu_{1}, \Delta' = \Delta_{1}, \varepsilon' = \varepsilon_{1}, T' = T_{1}$, $H/I$ has the unique weak Hopf algebra structure such that $\pi$ is a weak Hopf algebra morphism.

(3) Since $\pi$ is epimorphism, and $Ker\pi = I\subseteq Ker\varphi$, then we can prove the conclusion by the result of Theorem 3.2.

Theorem 3.5. (1) Let $A, B$ be weak Hopf algebras, and the weak antipodes $T_{A}$, $T_{B}$ of $A$ and $B$ are anti-algebra morphisms respectively. $\varphi: A\rightarrow B$ is a weak Hopf algebra morphism over the field $k$, then $\varphi^{\circ}:B^{\circ}\rightarrow A^{\circ}$ is also a weak Hopf algebra morphism.

(2) Let $\mathcal{C}$ be the category of the weak Hopf algebras which weak antipodes are anti-algebra morphisms, then, the anti-function from $\mathcal{C}$ to itself

is adjoint, that is to say that for any $A, B\in \mathcal{C}$, there is a natural isomorphism:

Proof. (1) At first, $\varphi^{\circ}$ is a bialgebra morphism on $k$. And, $\varphi T_{A} = T_{B}\varphi$, so $T_{A}^{*}\varphi^{\circ} = \varphi^{\circ}T_{B}^{*}$, that is $T_{A^{\circ}}\varphi^{\circ} = \varphi^{\circ}T_{B^{\circ}}$. Hence, $\varphi^{\circ}:B^{\circ}\rightarrow A^{\circ}$ is a weak Hopf algebra morphism.

(2) Let $\varphi: A\rightarrow B$ be a weak Hopf algebra morphism, then $\varphi^{\circ}:B^{\circ}\rightarrow A^{\circ}$ is a weak Hopf algebra morphism from the result of (1). Base on the inclusion relation $B^{\circ}\subset B^{*}$, we can deduce a k-algebra morphism $(B^{*})^{\circ}\rightarrow (B^{\circ})^{\circ}$. On the other hand, $B\subset (B^{*})^{\circ}$, so there is a coalgebra morphism

Since the identity of $(B^{\circ})^{\circ}$ is $\varepsilon(B^{\circ})$, then for any morphism $f\in B^{\circ}$, $f\mapsto f(1)$, this morphism is $\xi_{B}(1)$ exactly.

For any $a, b\in B$, $f\in B^{\circ}$,

That is $T_{(B^{\circ})^{\circ}}\circ \xi_{B} = \xi_{B}T_{B}$. Hence, $\xi_{B}$ is a weak Hopf algebra morphism.

At last, let $\varphi\in Hom_{weakHopf}(A, B^{\circ})$, $\psi\in Hom_{weakHopf}(B, A^{\circ})$, define

Obviously $\Phi(\theta(\varphi)) = \varphi$, $\theta(\Phi (\psi)) = \psi$, so $\theta$ is an isomorphism.

Let $S$ be a monoid with identity element $e$, then $kS$ is a weak Hopf algebra with $\Delta(x) = x\otimes x$, $\varepsilon(x) = e$, $T(x) = x^{*}$, $x^{*}\in V(x)$. The dual $k$-linear space $(kS)^{*}$ of $kS$ has $(kS)^{*}\cong Map(S, k)$, we denote $M_{k}(S) = Map(S, k)$. The following action makes $M_{k}(S)$ a two-sided $kS$-module.

For $f\in M_{k}(S)$, we denote the left $kS$-module, the right $kS$-module and the two-sided $kS$-module generated by $f$ respectively by $kSf$, $kfS$, and $kSfS$.

Theorem 3.6. ^[1] The $k$-linear map

is injective; the $k$-algebra morphism

has the following result:

Definition 3.3. ^[1] When $f\in M_{k}(S)$ satisfies the condition that $\mbox{dim}\ \ kSf < \infty$, $f$ is said to be a representative function on $S$. The set of all representative functions on $S$ is denoted by $R_{k}(S)$. $R_{k}(S)$ is a $k$-subalgebra of $M_{k}(S)$.

Now we have

Theorem 3.7. ^[1] $\delta (R_{k}(S))\subset \pi (R_{k}(S)\otimes R_{k}(S))$.

Thus, $\Delta = \pi^{-1}\circ \delta:R_{k}(S)\rightarrow R_{k}(S)\otimes R_{k}(S)$ is a $k$-algebra morphism that define a comultiplication on $R_{k}(S)$. Moreover, by defining a $k$-algebra morphism $\varepsilon: R_{k}(S)\rightarrow k$ by $\varepsilon(f) = f(e)$, $(R_{k}(S), \Delta, \varepsilon)$ becomes a $k$-coalgebra, and the fact that $\Delta$, $\varepsilon$ are $k$-algebra morphisms implies that $R_{k}(S)$ is a $k$-bialgebra. This is called the representative $k$-bialgebra of $S$.

Proposition 3.4. Let $S$ be a regular monoid, and $V(y)V(x)\subseteq V(xy)$ for any $x, y\in S$, then $R_{k}(S)$ becomes a commutative weak Hopf algebra.

Proof. Since $S$ is a regualr monoid, we can define

Then, we can prove $T(f)\in R_{k}(S)$ directly. Moreover,

Hence, $id*T*id = id$. Similarly, $T*id*T = T$. At the same time, $R_{k}(S)\subseteq M_{k}(S)$, so, $R_{k}(S)$ is a weak Hopf algebra.

Remark. When $S$ is an inverse monoid or commutative regular monoid, $V(y)V(x)\subseteq V(xy)$ is always right for any $x, y\in S$, then $R_{k}(S)$ will be a weak Hopf algebra from the above proposition.

Theorem 3.8. (1) Let $H$ be an affine $k$-algebra, if $S = M_{k}(H, k)$ is an affine algebra $k$-regular monoid, then $H$ has weak Hopf algebra structure.

(2) Let $H$ be a weak Hopf algebra, and an affine $k$-algebra as algebra, then $S = M_{k}(H, k)$ is an affine algebra $k$-regular monoid.

Proof. (1) Let $S = (M_{k}(H, k), w, m, t)$ be the affine $k$-regular monoid, then

are affine algebraic $k$-variety morphisms. Then, there are $k$-algebra morphisms $\Delta: H\rightarrow H\otimes H$, $T: H\rightarrow H$, $\varepsilon :H\rightarrow k$ such that

where $\lambda: M_{k}(H, k)\times M_{k}(H, k)\rightarrow M_{k}(H\otimes H)$ is an isomorphism, and $\lambda(\lambda\times id) = \lambda(id\times \lambda)$. Then for any $f, g, h\in M_{k}(H, k)$

Since $m(m\times id) = m(id\times m)$, $\lambda(\lambda\times id) = \lambda(id\times \lambda)$, and the arbitrariness of $f, g, h$, then $\Delta(\Delta\times id) = \Delta(id\times \Delta)$.

Since $m(f, e)(a) = f(a)$, hence for any $f\in M_{k}(H, k)$,

From the arbitrariness of $f$, $\sum_{(a)}a_{(1)}\varepsilon(a_{(2)}) = a$. Similarly, $\sum_{(a)} \varepsilon(a_{(1)})a_{(2)} = a$.

Moreover, $m(m\times id)(f, t(f), f)(a) = f(a)$, that is

hence, $\sum_{(a)}a_{(11)}T(a_{(12)})a_{(2)} = a$, that is to say $id*T*id = id$. Similarly, $T*id*T = T$.

In conclusion, $H$ is a weak Hopf algebra.

(2) Let $H = (H, \mu, \eta, \Delta, \varepsilon, T)$ be a weak Hopf algebra,

Then, $\lambda$ is an isomorphism.

Let

Then, for any $f, g, h\in M_{k}(H, k)$, $x\in H$,

Similarly,

Since $(\Delta\otimes id)\Delta = (id\otimes \Delta)\Delta$, so $m(m\times id) = m(id\times m)$.

So, $m(m\times id)(f, t(f), f) = f$. Similarly, $m(m\times id)(t(f), f, t(f)) = t(f)$.

At last,

that is $m(f, e) = f$. Similarly, $m(e, f) = f$. So, $S = M_{k}(H, k)$ is a regular monoid. Moreover, $H$ is an affine $k$-algebra, then $S = M_{k}(H, k)$ is an affine algebraic $k$-regular monoid.

Proof of Theorem 2.2. The theorem can get by the result of Proposition 2.4 and Theorem 3.8.

Example 3.1. Let $H = k[T, T']$ be a weak Hopf algebra which satisfies $TT'T = T$, $T'TT' = T'$, and

then $M_{k}(H, k)$ is an affine algebraic $k$-regular monoid.

Theorem 3.9. (1) Let $WH_{k}$ be the category of weak Hopf algebras, $MG_{r}$ be the category of regular monoids, for any $H, H_{1}, H_{2}\in ObjWH_{k}$, $g\in Hom_{WH_{k}}(H_{1}, H_{2})$, we can define

Then $\psi$ is an anti-function.

(2) Let $IG_{r}$ be the category of inverse mononids, for any $G$, $G_{1}$, $G_{2}\in ObjIG_{r}$, $f\in Hom_{IG_{r}}(G_{1}, G_{2})$, $\varphi_{2}\in R_{k}(G_{2})$, we can define

Then, $\phi$ is an anti-function too.

Proof. (1) For any $H_{1}, H_{2}, H_{3}\in ObjWH_{k}$, $g_{1}\in Hom_{WH_{k}}(H_{1}, H_{2})$, $g_{2}\in Hom_{WH_{k}}(H_{2}, H_{3})$. Based on the proof procedure of Theorem 3.8 (2), $\psi(H_{1})\in MG_{r}$. Next, We will prove that $\psi(g_{1})$ is a regular monoid morphism. Given $h\in H_{1}$, $\varphi_{1}, \varphi_{2}\in M_{k}(H_{1}, k)$,

On the other hand,

That is $\psi(g_{1})(\varphi_{2})*\psi(g_{1})(\varphi_{1}) = \psi(g_{1})(\varphi_{2}*\varphi_{1})$.

Moreover,

that is $\psi(g_{1})(e_{2}) = e_{1}$. Hence, $\psi(g_{1})$ is a regular monoid morphism.

Next, for any $h\in H_{1}$, $\varphi\in M_{k}(H_{1}, k)$,

Hence, $\psi(g_{1}\circ g_{2}) = \psi(g_{2})\circ \psi(g_{1})$. Similarly, we can prove $\psi(id_{H_{1}}) = id_{M_{k}(H_{1}, k)}$.

In conclusion, $\psi$ is an anti-function.

(2) Based on the remark of Proposition 3.4, $\phi(G_{1})\in ObjWH_{k}$. Next, we will prove that $\phi(f)$ is a weak Hopf algebra morphism. For any $a\in G_{1}$,

and

That is

Hence, $\phi(f)$ is an algebra morphism.

For any $a, a'\in G_{1}$,

where $\Delta_{R_{k}(G_{2})}\varphi_{2} = \sum_{(i)}r_{i}\otimes s_{i}$. So,

Moreover,

so, $\varepsilon_{R_{k}}(G_{1})\circ \phi(f) = \varepsilon_{R_{k}}(G_{2})$, that is $\phi(f)$ is a co-algebra morphism.

At last,

Hence, $\phi(f)T_{2} = T_{1}\phi(f)$.

In conclusion, $\phi(f)$ is a weak Hopf algebra morphism.

On the other side, for any $f_{1}\in Hom_{IG_{r}}(G_{1}, G_{2})$, $f_{2}\in Hom_{IG_{r}}(G_{2}, G_{3})$, $\varphi_{3}\in R_{k}(G_{3})$, $\varphi_{1}\in R_{k}(G_{1})$, we have

So, $\phi(f_{2}\circ f_{1}) = \phi(f_{1})\circ \phi(f_{2})$. Moreover,

that is $\phi(id_{G_{1}}) = id_{R_{k}(G_{1})}$.

In summary, $\phi$ is an anti-function.

5.   Conclusions

First of all, we have defined the affine algebraic regular monoid comparing with the concept of affine algebraic group and proved that the anti-function from the category $\mathcal{C}$ of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint at section 2. After that, we have characterized the affine weak k-algebra H whose affine k-variety S = Mk(H, k) admits a regular monoid structure: the category of affine algebraic regular monoids is anti-equivalent to the category of finitely generated commutative reduced weak Hopf algebras at section 3.

Acknowledgments

We would like to thank anonymous reviewers for their time and comments, and Prof. Fang Li of Zhejiang University whose suggestions and kind assistance are important for this article.

Conflict of interest

We declare that we have no conflict of interest.