Limits of Quantum B-Algebras

Abstract

Every set with a binary operation satisfying a true statement of propositional logic corresponds to a solution of the quantum Yang-Baxter equation. Quantum B-algebras and L-algebras are closely related to Yang-Baxter equation theory. In this paper, we study the categories with quantum B-algebras with morphisms of exact ones or spectral ones. We guarantee the existences of both direct limits and inverse limits.

1. Introduction and Preliminary

Every set with a binary operation satisfying a true statement of propositional logic corresponds to a solution of the quantum Yang-Baxter equation [1]. Quantum B-algebras [2] and L-algebras [3] are closely related to Yang-Baxter equation theory. In this paper, we consider direct limits and inverse limits in the categories with quantum B-algebras as objects, and with exact morphisms or spectral morphisms as morphisms. We show that both direct limits and inverse limits exist in these categories. Direct limits (dually inverse limits) are important topics in category theory, and have been studied deeply in some concrete categories, such as topological space category, abelian group category, commutative ring category and some logical algebra categories, etc (see [4,5,6,7,8]).

In this paper, the focus of attention will be on direct limits and inverse limits of quantum B-algebra categories. As a non-commutative generalization of locales, quantales were created in 1984 [9] to develop a framework for studying non-commutative spaces and quantum mechanics with a view toward non-commutative logics. Based on the implicational operators of quantales, Rump-Yang [2] introduced quantum B-algebras to unify logical algebras. As an important application of quantum B-algebras, Rump-Yang [10] generalized the classical fundamental theorem of arithmetic to non-commutative case. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales [2]. It is natural to consider the limits and the inverse limits of the categories of quantum B-algebras.

Recall that a quantum B-algebra [2] is a poset X with two binary operations → and ⇝, we write it by $X=(X;\to ,⇝,\le )$, such that the following conditions are satisfied: for all $x,y,z\in X$,

$$\begin{array}{c}\hfill y\to z\le (x\to y)\to (x\to z)\end{array}$$

(1)

$$\begin{array}{c}\hfill y⇝z\le (x⇝y)⇝(x⇝z)\end{array}$$

(2)

$$\begin{array}{c}\hfill y\le z⟹x\to y\le x\to z\end{array}$$

(3)

$$\begin{array}{c}\hfill x\le y\to z⟺y\le x⇝z.\end{array}$$

(4)

A quantum B-algebra $(X;\to ,⇝,\le )$ is called commutative if $x\to y=x⇝y$ holds for all $x,y\in X$. A quantum B-algebra X is called complete if $(X;\le )$ is a complete lattice (see [11]). X is called unital if there exists an element $u\in X$ such that $u\to x=u⇝x=x$ for all $x\in X$. Such an element u is unique if it exists and is called the unit of X [2].

We recall the following results for later applications.

Lemma 1

([2], Proposition 3). An algebra X with a partial order ≤ and two binary operations → and ⇝ is a quantum B-algebra if and only if it satisfies Equations (3) and (4) and

$$\begin{array}{c}\hfill x\to (y⇝z)=y⇝(x\to z)\end{array}$$

(5)

for all $x,y,z\in X$.

Lemma 2

([2]). Let $(X;\to ,⇝,\le )$ be a quantum B-algebra. Then for all $x,y,z\in X$, we have

1. 

$y\to z⟹(x⇝y)\to (x⇝z)$;

2. 

$y\le z⟹z\to x\le y\to x$;

3. 

$y\le z⟹z⇝x\le y⇝x$;

4. 

$x\le (x⇝y)\to y,x\le (x\to y)⇝y$;

5. 

$x\to y=\left(\right(x\to y)⇝y)\to y,x⇝y=\left(\right(x⇝y)\to y)⇝y$;

6. 

If u is the unit, then $u\le x⇝y\iff x\le y\iff u\le x\to y$;

7. 

If X has the smallest element 0, then $0\to 0=0⇝0$ is the greatest element (written it as 1), and $x\to 1=x⇝1=1$ for all $x\in X$;

8. 

If X is a lattice, then $(x\vee y)\to z=(x\to z)\wedge (y\to z)$ and $(x\vee y)⇝z=(x⇝z)\wedge (y⇝z)$.

Proposition 1 is the modification of Lemma 2, 8.

Proposition 1.

Let $(X;\to ,⇝,\le )$ be a complete quantum B-algebra, and Ω be an index set. Then $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\to z={\bigwedge }_{i\in \mathrm{\Omega }}(x_i\to z)$ and $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)⇝z={\bigwedge }_{i\in \mathrm{\Omega }}(x_i⇝z)$ for all $x_i(i\in \mathrm{\Omega }),z\in X$.

Proof. 

Assume that $(X;\to ,⇝,\le )$ is a complete quantum B-algebra. Let $x_i(i\in \mathrm{\Omega }),z\in X$. For each $j\in \mathrm{\Omega }$, since $x_j\le {\bigvee }_{i\in \mathrm{\Omega }}{x}_i$, we have $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\to z\le x_j\to z$ by Lemma 2, 2, and so $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\to z\le {\bigwedge }_{i\in \mathrm{\Omega }}(x_i\to z)$. Also, if $y\le x_j\to z$ for each $j\in \mathrm{\Omega }$, then $x_j\le y⇝z$ by Equation (4). It follows that $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\le y⇝z$, and so $y\le \left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\to z$ by Equation (4) again. Therefore, $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)\to z={\bigwedge }_{i\in \mathrm{\Omega }}(x_i\to z)$.

Similarly, we can show that $\left({\bigvee }_{i\in \mathrm{\Omega }}{x}_i\right)⇝z={\bigwedge }_{i\in \mathrm{\Omega }}(x_i⇝z)$. □

Let $(X;{\to }_X,{⇝}_X,{\le }_X)$ and $(Y;{\to }_Y,{⇝}_Y,{\le }_Y)$ be quantum B-algebras. Recall that a morphism $f:X\to Y$ of quantum B-algebras is a monotonic map (that is to say, for all $a,b\in X$, $a{\le }_{X}b$ implies that $f\left(a\right){\le }_{Y}f\left(b\right))$ which satisfies the inequalities

$$\begin{array}{c}\hfill f\left(a{\to }_{X}b\right){\le }_{Y}f\left(a\right){\to }_{Y}f\left(b\right),f\left(a{⇝}_{X}b\right){\le }_{Y}f\left(a\right){⇝}_{Y}f\left(b\right)\end{array}$$

(6)

for all $a,b\in X$. A morphism $f:X\to Y$ of quantum B-algebras is called injective (resp. surjective, isomorphic) if f is an injective (resp. surjective, bijective) map; f is called exact if the inequalities (6) are equations; and f is called spectral if f is exact such that for all $y\in Y$ and $z\in f\left(X\right)$, the element $y\to z$ belongs to $f\left(X\right)$. In short: $Y\to f\left(X\right)\subseteq f\left(X\right)$ [2].

The category of quantum B-algebras will be denoted by $qBAlg$. In [2], Rump-Yang shows that the quantum B-algebras with spectral morphisms form a subcategory $qB$ of $qBAlg$. Evidently, the quantum B-algebras with exact morphisms also form a subcategory $qBE$ of $qBAlg$, and $qB$ is a subcategory of $qBE$.

Let both X and Y be unital quantum B-algebras, and $u_X$ and $u_Y$ be, respectively, the unit element of X and Y. A morphism $f:X\to Y$ of unital quantum B-algebras is called unital if f is exact and $f\left(u_X\right)=u_Y$. It is easy to see that the unital quantum B-algebras with unital morphisms form a subcategory $qBU$ of $qBE$.

In this paper, we will study direct limits and inverse limits of the categories of quantum B-algebras. Let $\mathcal{D}$ be a category such that $\mathcal{D}\in \{qBE,qB,qBU\}$. In Section 2 we prove the existence of direct limits for a directed system in the category $\mathcal{D}$ (Theorem 1 and Corollary 1). In Section 3, we obtain the existence of inverse limits for a inverse system in the category $\mathcal{D}$ (Theorem 2 and Proposition 2).

2. Direct Limits of Quantum $\mathit{B}$-Algebras

Throughout what follows, $\mathcal{C}$ will denote an arbitrary category; $A\in ob\mathcal{C}$ means that A is an object of $\mathcal{C}$; for any $A,B\in ob\mathcal{C}$, put

$$Ho{m}_{\mathcal{C}}(A,B)=\left\{f\right|f:A\to B \mathrm{is} \mathrm{a} \mathrm{morphism} \mathrm{in} \mathcal{C}\}.$$

We should bear in mind that $qBE($resp. $qB)$ denotes the category with quantum B-algebras as objects and exact (resp. spectral) morphisms as morphisms; and $qBU$ denotes the category with unital quantum B-algebras as objects and unital morphisms as morphisms.

Recall that a partially ordered set $(D,\le )$ is called an upward directed set if for any $\lambda ,\mu \in D$, there exists an element $\nu \in D$ such that $\lambda \le \nu$ and $\mu \le \nu$.

Definition 1

([12]). A directed system in $\mathcal{C}$ is a pair ${(A_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ satisfying the following conditions:

1. 

$(D,\le )$ is an upward directed set.

2. 

$A_{\lambda }\in ob\mathcal{C}$ for each $\lambda \in D$.

3. 

$A_{\lambda }\cap A_{\mu }=\varnothing$ if $\lambda \ne \mu$ in D.

4. 

${\varphi }_{\lambda \mu }\in Ho{m}_{\mathcal{C}}(A_{\lambda },A_{\mu })$ whenever $\lambda \le \mu$ in D.

5. 

${\varphi }_{\mu \nu }{\varphi }_{\lambda \mu }={\varphi }_{\lambda \nu }$ whenever $\lambda \le \mu \le \nu$ in D.

6. 

${\varphi }_{\lambda \lambda }=i{d}_{A_{\lambda }}$, where $i{d}_{A_{\lambda }}$ is the identity map of $A_{\lambda }$ for all $\lambda \in D$.

Thus if $\lambda \le \mu \le \nu$ and $\lambda \le \delta \le \nu$ in D, then ${\varphi }_{\mu \nu }{\varphi }_{\lambda \mu }={\varphi }_{\delta \nu }{\varphi }_{\lambda \delta }$.

Definition 2

([5]). A direct limit for a directed system ${(A_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ in $\mathcal{C}$ is a pair $(A;{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$, where $A\in ob\mathcal{C}$ and each ${\varphi }_{\lambda }\in Ho{m}_{\mathcal{C}}(A_{\lambda },A)$, such that

1. 

${\varphi }_{\mu }{\varphi }_{\lambda \mu }={\varphi }_{\lambda }$ whenever $\lambda \le \mu$ in D.

2. 

For any $B\in ob\mathcal{C}$, if ${\sigma }_{\mu }\in Ho{m}_{\mathcal{C}}(A_{\mu },B)$ such that ${\sigma }_{\mu }{\varphi }_{\lambda \mu }={\sigma }_{\lambda }$ whenever $\lambda \le \mu$ in D, then there exists a unique $\sigma \in Ho{m}_{\mathcal{C}}(A,B)$ such that ${\sigma }_{\mu }=\sigma {\varphi }_{\mu }$ for all $\mu \in D$.

By Definition 2, we know that if both $(A;{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ and $(B;{\left\{{\sigma }_{\lambda }\right\}}_{\lambda \in D})$ are direct limits for a directed system ${(A_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ in $\mathcal{C}$, then A is isomorphic to B in $\mathcal{C}$.

In what follows, unless otherwise specified, we always assume that D is an upward directed set, and $X_{\lambda }=(X_{\lambda };{\to }_{\lambda },{⇝}_{\lambda },{\le }_{\lambda })$ is a quantum B-algebra for every $\lambda \in D$.

Theorem 1.

If ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ is a directed system in the category $qBE$ (resp. $qB$), then direct limits for ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ exist in the category $qBE$ (resp. $qB$).

Moreover, if $X_{\lambda }$ is commutative for every $\lambda \in D$, then direct limits for ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ are also commutative.

Proof. 

Assume that ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ is a directed system in the category $qBE$. Put $X={\bigcup }_{\lambda \in D}{X}_{\lambda }$, and define a binary relation ≡ on X as follows: for any $a\in X_{\lambda }$ and $b\in X_{\mu }$,

$$a\equiv b⟺(\exists \nu \in D,\nu \ge \lambda ,\mu ){\varphi }_{\lambda \nu }\left(a\right)={\varphi }_{\mu \nu }\left(b\right).$$

Then ≡ is an equivalence relation. In fact, it is clear that ≡ is reflexive and symmetric. To prove it also satisfies transitivity, suppose that $a\equiv b$ and $b\equiv c$, where $a\in X_{\lambda }$, $b\in X_{\mu },c\in X_{\kappa }$. Then there exist $\nu ,\omega \in D$ such that $\nu \ge \lambda ,\mu$, ${\varphi }_{\lambda \nu }\left(a\right)={\varphi }_{\mu \nu }\left(b\right)$; and $\omega \ge \mu ,\kappa$, ${\varphi }_{\mu \omega }\left(b\right)={\varphi }_{\kappa \omega }\left(c\right)$. Since D is upward directed, there exists $\gamma \in D$ such that $\gamma \ge \nu ,\omega$. It follows that

$${\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\nu \gamma }{\varphi }_{\lambda \nu }\left(a\right)={\varphi }_{\nu \gamma }{\varphi }_{\mu \nu }\left(b\right)={\varphi }_{\omega \gamma }{\varphi }_{\mu \omega }\left(b\right)={\varphi }_{\omega \gamma }{\varphi }_{\kappa \omega }\left(c\right)={\varphi }_{\kappa \gamma }\left(c\right),$$

and hence $a\equiv c$.

Put $\overline{a}=\{b\in X|b\equiv a\}$ and $\overline{X}=\left\{\overline{a}\right|a\in X\}$. Define a binary relation ⪯ on $\overline{X}$ by: for any $a\in X_{\lambda },b\in X_{\mu }$,

$$\overline{a}⪯\overline{b}⟺(\exists \nu \in D,\nu \ge \lambda ,\mu ){\varphi }_{\lambda \nu }\left(a\right){\le }_{\nu }{\varphi }_{\mu \nu }\left(b\right).$$

Then ⪯ is well defined. Indeed, suppose that $a\in X_{\lambda },b\in X_{\mu }$, $c\in X_{\alpha },d\in X_{\beta }$ such that $\overline{a}=\overline{c}$, $\overline{b}=\overline{d}$ and $\overline{a}⪯\overline{b}$. Then there exist $\gamma ,\eta ,\nu \in D$ such that $\gamma \ge \alpha ,\lambda$ with ${\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\alpha \gamma }\left(c\right)$; and $\eta \ge \mu ,\beta$ with ${\varphi }_{\mu \eta }\left(b\right)={\varphi }_{\beta \eta }\left(d\right)$; and $\nu \ge \lambda ,\mu$ with ${\varphi }_{\lambda \nu }\left(a\right){\le }_{\nu }{\varphi }_{\mu \nu }\left(b\right)$. Since D is an upward directed set, there exists $\delta \in D$ such that $\delta \ge \gamma ,\eta ,\nu$. It follows that

$$\begin{array}{cc}\hfill {\varphi }_{\alpha \delta }\left(c\right)& ={\varphi }_{\gamma \delta }{\varphi }_{\alpha \gamma }\left(c\right)={\varphi }_{\gamma \delta }{\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\nu \delta }{\varphi }_{\lambda \nu }\left(a\right)\hfill \\ \hfill & {\le }_{\delta }{\varphi }_{\nu \delta }{\varphi }_{\mu \nu }\left(b\right)={\varphi }_{\eta \delta }{\varphi }_{\mu \eta }\left(b\right)={\varphi }_{\eta \delta }{\varphi }_{\beta \eta }\left(d\right)={\varphi }_{\beta \delta }\left(d\right).\hfill \end{array}$$

and so $\overline{c}⪯\overline{d}.$ Hence ⪯ is well defined.

Now we shall show that ⪯ is a partial order on $\overline{X}$. In fact, it is clear that ⪯ is reflexive. To prove that ⪯ is antisymmetric, let $a\in X_{\lambda },b\in X_{\mu }$ such that $\overline{a}⪯\overline{b}$ and $\overline{b}⪯\overline{a}$. Then there exist $\nu ,\omega \in D$ such that $\nu \ge \lambda ,\mu$ with ${\varphi }_{\lambda \nu }\left(a\right){\le }_{\nu }{\varphi }_{\mu \nu }\left(b\right)$; and $\omega \ge \lambda ,\mu$ with ${\varphi }_{\mu \omega }\left(b\right){\le }_{\omega }{\varphi }_{\lambda \omega }\left(a\right)$. So for any $\gamma \in D$ with $\gamma \ge \nu ,\omega$, we have

$$\begin{gathered} {\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\nu \gamma }{\varphi }_{\lambda \nu }\left(a\right){\le }_{\gamma }{\varphi }_{\nu \gamma }{\varphi }_{\mu \nu }\left(b\right)={\varphi }_{\mu \gamma }\left(b\right) \mathrm{and} \\ {\varphi }_{\mu \gamma }\left(b\right)={\varphi }_{\omega \gamma }{\varphi }_{\mu \omega }\left(b\right){\le }_{\gamma }{\varphi }_{\omega \gamma }{\varphi }_{\lambda \omega }\left(a\right)={\varphi }_{\lambda \gamma }\left(a\right). \end{gathered}$$

It follows that ${\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\mu \gamma }\left(b\right)$, and so $\overline{a}=\overline{b}$. Thus ⪯ is antisymmetric.

To prove that ⪯ also satisfies transitivity, assume that $\overline{a}⪯\overline{b}$ and $\overline{b}⪯\overline{c}$, where $a\in X_{\lambda }$, $b\in X_{\mu }$, $c\in X_{\kappa }$. Then there exist $\nu ,\omega \in D$ such that $\nu \ge \lambda ,\mu$ with ${\varphi }_{\lambda \nu }\left(a\right){\le }_{\nu }{\varphi }_{\mu \nu }\left(b\right)$; and $\omega \ge \mu ,\kappa$ with ${\varphi }_{\mu \omega }\left(b\right){\le }_{\omega }{\varphi }_{\kappa \omega }\left(c\right)$. So for any $\gamma \in D$ with $\gamma \ge \nu ,\omega$, we have

$$\begin{gathered} {\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\nu \gamma }{\varphi }_{\lambda \nu }\left(a\right){\le }_{\gamma }{\varphi }_{\nu \gamma }{\varphi }_{\mu \nu }\left(b\right)={\varphi }_{\mu \gamma }\left(b\right) \mathrm{and} \\ {\varphi }_{\mu \gamma }\left(b\right)={\varphi }_{\omega \gamma }{\varphi }_{\mu \omega }\left(b\right){\le }_{\gamma }{\varphi }_{\omega \gamma }{\varphi }_{\kappa \omega }\left(c\right)={\varphi }_{\kappa \gamma }\left(c\right). \end{gathered}$$

It follows that ${\varphi }_{\lambda \gamma }\left(a\right){\le }_{\gamma }{\varphi }_{\kappa \gamma }\left(c\right)$, and so $\overline{a}⪯\overline{c}$. Hence ⪯ satisfies transitivity, and thus ⪯ is a partial order on $\overline{X}$.

To define operations on $\overline{X}$ such that $\overline{X}$ becomes a quantum B-algebra, we first establish the following result.

Claim $(⧫)$: if $a\in X_{\lambda },b\in X_{\mu }$, then for any $\nu ,\omega \in D$ such that $\nu \ge \lambda ,\mu$ and $\omega \ge \lambda ,\mu$, we have

$$\begin{array}{c}\hfill \overline{{\varphi }_{\lambda \nu }\left(a\right){\to }_{\nu }{\varphi }_{\mu \nu }\left(b\right)}=\overline{{\varphi }_{\lambda \omega }\left(a\right){\to }_{\omega }{\varphi }_{\mu \omega }\left(b\right)}\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \overline{{\varphi }_{\lambda \nu }\left(a\right){⇝}_{\nu }{\varphi }_{\mu \nu }\left(b\right)}=\overline{{\varphi }_{\lambda \omega }\left(a\right){⇝}_{\omega }{\varphi }_{\mu \omega }\left(b\right)}.\end{array}$$

(8)

Indeed, since D is an upward directed set, there exists $\gamma \in D$ such that $\gamma \ge \nu ,\omega$. Notice that both ${\varphi }_{\nu \gamma }$ and ${\varphi }_{\omega \gamma }$ are exact morphisms of quantum B-algebras, we have

$$\begin{array}{ccc}\hfill {\varphi }_{\nu \gamma }\left({\varphi }_{\lambda \nu }\left(a\right){\to }_{\nu }{\varphi }_{\mu \nu }\left(b\right)\right)& =& {\varphi }_{\nu \gamma }{\varphi }_{\lambda \nu }\left(a\right){\to }_{\gamma }{\varphi }_{\nu \gamma }{\varphi }_{\mu \nu }\left(b\right)\hfill \\ & =& {\varphi }_{\omega \gamma }{\varphi }_{\lambda \omega }\left(a\right){\to }_{\gamma }{\varphi }_{\omega \gamma }{\varphi }_{\mu \omega }\left(b\right)\hfill \\ & =& {\varphi }_{\omega \gamma }\left({\varphi }_{\lambda \omega }\left(a\right){\to }_{\omega }{\varphi }_{\mu \omega }\left(b\right)\right),\hfill \end{array}$$

and so Equation (7) holds. Similarly, we can verify that Equation (8) holds.

Now, define two binary operations → and ⇝ on $\overline{X}$ by:

$$\overline{a}\to \overline{b}=\overline{{\varphi }_{\lambda \nu }\left(a\right){\to }_{\nu }{\varphi }_{\mu \nu }\left(b\right)} \mathrm{and} \overline{a}⇝\overline{b}=\overline{{\varphi }_{\lambda \nu }\left(a\right){⇝}_{\nu }{\varphi }_{\mu \nu }\left(b\right)},$$

where $a\in X_{\lambda },b\in X_{\mu }$, and $\nu \in D$ with $\nu \ge \lambda ,\mu$. To prove that the operation → is well defined, let $a\in X_{\lambda },b\in X_{\mu }$, $c\in X_{\alpha },d\in X_{\beta }$ such that $\overline{a}=\overline{c}$ and $\overline{b}=\overline{d}$. Then there exist $\gamma ,\eta \in D$ such that $\gamma \ge \alpha ,\lambda$ with ${\varphi }_{\lambda \gamma }\left(a\right)={\varphi }_{\alpha \gamma }\left(c\right)$; and $\eta \ge \mu ,\beta$ with ${\varphi }_{\mu \eta }\left(b\right)={\varphi }_{\beta \eta }\left(d\right)$. Since D is an upward directed set, there exists $\kappa \in D$ such that $\kappa \ge \gamma ,\eta$. By Claim $(⧫)$, we have

$$\begin{array}{ccc}\hfill \overline{a}\to \overline{b}& =& \overline{{\varphi }_{\lambda \kappa }\left(a\right){\to }_{\kappa }{\varphi }_{\mu \kappa }\left(b\right)}\hfill \\ & =& \overline{{\varphi }_{\gamma \kappa }{\varphi }_{\lambda \gamma }\left(a\right){\to }_{\kappa }{\varphi }_{\eta \kappa }{\varphi }_{\mu \eta }\left(b\right)}\hfill \\ & =& \overline{{\varphi }_{\gamma \kappa }{\varphi }_{\alpha \gamma }\left(c\right){\to }_{\kappa }{\varphi }_{\eta \kappa }{\varphi }_{\beta \eta }\left(d\right)}\hfill \\ & =& \overline{{\varphi }_{\alpha \kappa }\left(c\right){\to }_{\kappa }{\varphi }_{\beta \kappa }\left(d\right)}\hfill \\ & =& \overline{c}\to \overline{d}.\hfill \end{array}$$

Thus the operation → is well defined. Similarly, we can prove that the operation ⇝ is also well defined.

Next, we will show that $(\overline{X};\to ,⇝,⪯)$ is a quantum B-algebra. In fact, let $x\in X_{\lambda },y\in X_{\mu }$ and $z\in X_{\nu }$. We have

(i) If $\overline{y}⪯\overline{z}$, then ${\varphi }_{\mu \omega }\left(y\right){\le }_{\omega }{\varphi }_{\nu \omega }\left(z\right)$ for some $\omega \in D$ with $\omega \ge \mu ,\nu$. Since D is an upward directed set, there is $\kappa \in D$ such that $\kappa \ge \omega ,\lambda$. Notice that ${\varphi }_{\mu \kappa }:X_{\mu }\to X_{\kappa }$ is an exact homomorphism of quantum B-algebras, Since $X_{\kappa }$ is a quantum B-algebra. So, by definitions of ⪯, →, and Equation (1) we get

$$\begin{array}{ccc}& & {\varphi }_{\mu \omega }\left(y\right){\le }_{\omega }{\varphi }_{\nu \omega }\left(z\right)\hfill \\ & \Rightarrow & {\varphi }_{\mu \kappa }\left(y\right)={\varphi }_{\omega \kappa }{\varphi }_{\mu \omega }\left(y\right){\le }_{\kappa }{\varphi }_{\omega \kappa }{\varphi }_{\nu \omega }\left(z\right)={\varphi }_{\nu \kappa }\left(z\right)\hfill \\ & \Rightarrow & {\varphi }_{\lambda \kappa }\left(x\right){\to }_{\kappa }{\varphi }_{\mu \kappa }\left(y\right){\le }_{\kappa }{\varphi }_{\lambda \kappa }\left(x\right){\to }_{\kappa }{\varphi }_{\nu \kappa }\left(z\right)\hfill \\ & \Rightarrow & \overline{{\varphi }_{\lambda \kappa }\left(x\right){\to }_{\kappa }{\varphi }_{\mu \kappa }\left(y\right)}⪯\overline{{\varphi }_{\lambda \kappa }\left(x\right){\to }_{\kappa }{\varphi }_{\nu \kappa }\left(z\right)}\hfill \\ & \Rightarrow & \overline{x}\to \overline{y}⪯\overline{x}\to \overline{z}\hfill \end{array}$$

Thus $(\overline{X};\to ,⇝,⪯)$ satisfies Equation (3).

(ii) If $\overline{x}⪯\overline{y}\to \overline{z}$, then by the definition of →, we have $\overline{x}⪯\overline{{\varphi }_{\mu \omega }\left(y\right){\to }_{\omega }{\varphi }_{\nu \omega }\left(z\right)}$ for some $\omega \in D$ with $\omega \ge \mu ,\nu$. It follows by the definition of ⪯ that there exists $\gamma \in D$ with $\gamma \ge \lambda ,\omega$ such that

$${\varphi }_{\lambda \gamma }\left(x\right){\le }_{\gamma }{\varphi }_{\omega \gamma }\left({\varphi }_{\mu \omega }\left(y\right){\to }_{\omega }{\varphi }_{\nu \omega }\left(z\right)\right)={\varphi }_{\mu \gamma }\left(y\right){\to }_{\gamma }{\varphi }_{\nu \gamma }\left(z\right).$$

Since $X_{\gamma }$ is a quantum B-algebra, we have ${\varphi }_{\mu \gamma }\left(y\right){\le }_{\gamma }{\varphi }_{\lambda \gamma }\left(x\right){⇝}_{\gamma }{\varphi }_{\nu \gamma }\left(z\right)$ by Equation (4), and so

$$\overline{y}=\overline{{\varphi }_{\mu \gamma }\left(y\right)}⪯\overline{{\varphi }_{\lambda \gamma }\left(x\right){⇝}_{\gamma }{\varphi }_{\nu \gamma }\left(z\right)}=\overline{x}⇝\overline{z}$$

by the definitions of ⪯ and ⇝. Similarly, we can show that $\overline{y}⪯\overline{x}⇝\overline{z}$ implies that $\overline{x}⪯\overline{y}\to \overline{z}$, and thus $(\overline{X};\to ,⇝,⪯)$ satisfies Equation (4).

(iii) Since D is an upward directed set, there exists $\gamma \in D$ such that $\gamma \ge \lambda ,\mu ,\nu$. By Claim $(⧫)$, and since $X_{\gamma }$ is a quantum B-algebra and by Equation (5) we obtain

$$\begin{array}{ccc}& & \overline{x}\to (\overline{y}⇝\overline{z})\hfill \\ & =& \overline{x}\to (\overline{{\varphi }_{\mu \gamma }\left(y\right){⇝}_{\gamma }{\varphi }_{\nu \gamma }\left(z\right)})\hfill \\ & =& \overline{{\varphi }_{\lambda \gamma }\left(x\right){\to }_{\gamma }\left({\varphi }_{\mu \gamma }\left(y\right){⇝}_{\gamma }{\varphi }_{\nu \gamma }\left(z\right)\right)}\hfill \\ & =& \overline{{\varphi }_{\mu \gamma }\left(y\right){⇝}_{\gamma }\left({\varphi }_{\lambda \gamma }\left(x\right){\to }_{\gamma }{\varphi }_{\nu \gamma }\left(z\right)\right)}\hfill \\ & =& \overline{y}⇝(\overline{x}\to \overline{z}),\hfill \end{array}$$

that is to say, $(\overline{X};\to ,⇝,⪯)$ satisfies Equation (5), and therefore $(\overline{X};\to ,⇝,⪯)$ is a quantum B-algebra by Lemma 1.

For any $\lambda \in D$, define ${\varphi }_{\lambda }:X_{\lambda }\to \overline{X}$ by ${\varphi }_{\lambda }\left(x\right)=\overline{x}$, where $x\in X_{\lambda }$. We claim that $(\overline{X};{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ is a direct limit of the directed system ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ in the category $qBE$. In fact, firstly, for any $a,b\in X_{\lambda }$, we have

$${\varphi }_{\lambda }\left(a{\to }_{\lambda }b\right)=\overline{a{\to }_{\lambda }b}=\overline{a}\to \overline{b}={\varphi }_{\lambda }\left(a\right)\to {\varphi }_{\lambda }\left(b\right),$$

$${\varphi }_{\lambda }\left(a{⇝}_{\lambda }b\right)=\overline{a{⇝}_{\lambda }b}=\overline{a}⇝\overline{b}={\varphi }_{\lambda }\left(a\right)⇝{\varphi }_{\lambda }\left(b\right),$$

and

$$a{\le }_{\lambda }b⟹{\varphi }_{\lambda }\left(a\right)=\overline{a}⪯\overline{b}={\varphi }_{\lambda }\left(b\right)$$

by the definition of ⪯. Thus ${\varphi }_{\lambda }$ is an exact morphism of quantum B-algebras.

Secondly, let $\lambda \le \mu$ in D. For any $x\in X_{\lambda }$, notice that ${\varphi }_{\lambda \mu }\left(x\right)\equiv x$ since ${\varphi }_{\mu \mu }\left({\varphi }_{\lambda \mu }\left(x\right)\right)={\varphi }_{\lambda \mu }\left(x\right)$, we have ${\varphi }_{\mu }\left({\varphi }_{\lambda \mu }\left(x\right)\right)=\overline{{\varphi }_{\lambda \mu }\left(x\right)}=\overline{x}={\varphi }_{\lambda }\left(x\right)$, so ${\varphi }_{\mu }{\varphi }_{\lambda \mu }={\varphi }_{\lambda }$, and hence $(\overline{X};{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ satisfies the Condition 1 of Definition 2.

Finally, to prove $(\overline{X};{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ also satisfies the Condition 2 of Definition 2, suppose that $Y=(Y;{\to }_Y,{⇝}_Y,{\le }_Y)$ is a quantum B-algebra and ${\sigma }_{\mu }:X_{\mu }\to Y$ is an exact morphism such that ${\sigma }_{\mu }{\varphi }_{\lambda \mu }={\sigma }_{\lambda }$ whenever $\lambda \le \mu$ in D.

Define a function $\sigma :\overline{X}\to Y;\overline{x}\mapsto {\sigma }_{\lambda }\left(x\right)$ if $x\in X_{\lambda }$. Then $\sigma$ is well defined. Indeed, if $\overline{a}=\overline{b}$, where $a\in X_{\alpha }$ and $b\in X_{\beta }$, then there exists $\gamma \in D$ such that $\gamma \ge \alpha ,\beta$ and ${\varphi }_{\alpha \gamma }\left(a\right)={\varphi }_{\beta \gamma }\left(b\right)$, which implies that

$$\sigma \left(\overline{a}\right)={\sigma }_{\alpha }\left(a\right)={\sigma }_{\gamma }{\varphi }_{\alpha \gamma }\left(a\right)={\sigma }_{\gamma }{\varphi }_{\beta \gamma }\left(b\right)={\sigma }_{\beta }\left(b\right)=\sigma \left(\overline{b}\right),$$

and hence $\sigma$ is well defined.

We now claim that $\sigma$ is an exact morphism of quantum B-algebras. Indeed, for any $\overline{x},\overline{y}\in \overline{X}$, where $x\in X_{\lambda },y\in X_{\mu }$. Since D is an upward directed set, there exists $\nu \in D$ such that $\nu \ge \lambda ,\mu$. Since ${\sigma }_{\nu }:X_{\nu }\to Y$ is an exact morphism. So, by definitions of →, ⇝, and $\sigma$ we have

$$\begin{array}{ccc}\hfill \sigma (\overline{x}\to \overline{y})& =& \sigma (\overline{{\varphi }_{\lambda \nu }\left(x\right){\to }_{\nu }{\varphi }_{\mu \nu }\left(y\right)})\hfill \\ & =& {\sigma }_{\nu }\left({\varphi }_{\lambda \nu }\left(x\right){\to }_{\nu }{\varphi }_{\mu \nu }\left(y\right)\right)\hfill \\ & =& {\sigma }_{\nu }{\varphi }_{\lambda \nu }\left(x\right){\to }_Y{\sigma }_{\nu }{\varphi }_{\mu \nu }\left(y\right)\hfill \\ & =& {\sigma }_{\lambda }\left(x\right){\to }_Y{\sigma }_{\mu }\left(y\right)\hfill \\ & =& \sigma \left(\overline{x}\right){\to }_Y\sigma \left(\overline{y}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \sigma (\overline{x}⇝\overline{y})& =& \sigma (\overline{{\varphi }_{\lambda \nu }\left(x\right){⇝}_{\nu }{\varphi }_{\mu \nu }\left(y\right)})\hfill \\ & =& {\sigma }_{\nu }\left({\varphi }_{\lambda \nu }\left(x\right){⇝}_{\nu }{\varphi }_{\mu \nu }\left(y\right)\right)\hfill \\ & =& {\sigma }_{\nu }{\varphi }_{\lambda \nu }\left(x\right){⇝}_Y{\sigma }_{\nu }{\varphi }_{\mu \nu }\left(y\right)\hfill \\ & =& {\sigma }_{\lambda }\left(x\right){⇝}_Y{\sigma }_{\mu }\left(y\right)\hfill \\ & =& \sigma \left(\overline{x}\right){⇝}_Y\sigma \left(\overline{y}\right).\hfill \end{array}$$

Moreover, if $\overline{x}⪯\overline{y}$, where $x\in X_{\lambda },y\in X_{\mu }$, then there exists $\omega \in D$ such that $\omega \ge \lambda ,\mu$ and ${\varphi }_{\lambda \omega }\left(x\right){\le }_{\omega }{\varphi }_{\mu \omega }\left(y\right)$. Since ${\sigma }_{\omega }:X_{\omega }\to Y$ is an exact morphism of quantum B-algebras, we immediately obtain that

$$\sigma \left(\overline{x}\right)={\sigma }_{\lambda }\left(x\right)={\sigma }_{\omega }\left({\varphi }_{\lambda \omega }\left(x\right)\right){\le }_Y{\sigma }_{\omega }\left({\varphi }_{\mu \omega }\left(y\right)\right)={\sigma }_{\mu }\left(y\right)=\sigma \left(\overline{y}\right).$$

Therefore $\sigma :\overline{X}\to Y$ is an exact morphism of quantum B-algebras.

Also, for any $\mu \in D$, we have $\sigma {\varphi }_{\mu }={\sigma }_{\mu }$, since $\sigma {\varphi }_{\mu }\left(x\right)=\sigma \left({\varphi }_{\mu }\left(x\right)\right)=\sigma \left(\overline{x}\right)={\sigma }_{\mu }\left(x\right)$ for any $x\in X_{\mu }$. To prove that such $\sigma$ is unique, suppose that $\tau :\overline{X}\to Y$ is an exact morphism of quantum B-algebras such that $\tau {\varphi }_{{}_{\mu }}={\sigma }_{\mu }$ for all $\mu \in D$. Then for any $\overline{a}\in \overline{X}$, where $a\in X_{\alpha }$, we have $\tau \left(\overline{a}\right)=\tau \left({\varphi }_{\alpha }\left(a\right)\right)=\tau {\varphi }_{\alpha }\left(a\right)={\sigma }_{\alpha }\left(a\right)=\sigma \left(\overline{a}\right)$, and so $\tau =\sigma$. Thus such $\sigma$ is unique, and so $(\overline{X};{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ satisfies the Condition 2 of Definition 2. Therefore $(\overline{X};{\left\{{\varphi }_{\lambda }\right\}}_{\lambda \in D})$ is a direct limit of the directed system ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ in the category $qBE$.

Moreover, if each $X_{\lambda }$ is a commutative quantum B-algebra, then it is easy to see that $\overline{X}$ is also a commutative quantum B-algebra, and hence direct limits for ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ are commutative.

Finally, to prove direct limits for any directed system in the category $qB$ exist, suppose that ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ is a directed system in the category $qB$. It suffices to show that following both $\left({▸}_1\right)$ and $\left({▸}_2\right)$ hold.

$\left({▸}_1\right)$ The ${\varphi }_{\lambda }$ which is defined above (${\varphi }_{\lambda }:X_{\lambda }\to \overline{X};x\mapsto \overline{x}$ for any $x\in X_{\lambda }$) is a spectral morphism of quantum B-algebras for each $\lambda \in D$.

Indeed, suppose that $b\in X_{\lambda }$ and $\overline{a}\in \overline{X}$, where $a\in X_{\alpha }$. Let $\mu \in D$ such that $\mu \ge \lambda ,\alpha$. Since ${\varphi }_{\lambda \mu }:X_{\lambda }\to X_{\mu }$ is a spectral morphism, noticing that ${\varphi }_{\alpha \mu }\left(a\right)\in X_{\mu }$ and ${\varphi }_{\lambda \mu }\left(b\right)\in {\varphi }_{\lambda \mu }\left(X_{\lambda }\right)$, we get ${\varphi }_{\alpha \mu }\left(a\right){\to }_{\mu }{\varphi }_{\lambda \mu }\left(b\right)$ belongs to ${\varphi }_{\lambda \mu }\left(X_{\lambda }\right)$, and so ${\varphi }_{\alpha \mu }\left(a\right){\to }_{\mu }{\varphi }_{\lambda \mu }\left(b\right)={\varphi }_{\lambda \mu }\left(c\right)$ for some $c\in X_{\lambda }$. Since ${\varphi }_{\lambda \mu }\left(c\right)\equiv c$, it follows that

$$\overline{a}\to {\varphi }_{\lambda }\left(b\right)=\overline{a}\to \overline{b}=\overline{{\varphi }_{\alpha \mu }\left(a\right){\to }_{\mu }{\varphi }_{\lambda \mu }\left(b\right)}=\overline{{\varphi }_{\lambda \mu }\left(c\right)}=\overline{c}={\varphi }_{\lambda }\left(c\right)\in {\varphi }_{\lambda }\left(X_{\lambda }\right),$$

and hence ${\varphi }_{\lambda }$ is a spectral morphism of quantum B-algebras.

$\left({▸}_2\right)$ If ${\{{\sigma }_{\mu }:X_{\mu }\to Y\}}_{\mu \in D}$ is a family of spectral morphisms of quantum B-algebras such that ${\sigma }_{\mu }{\varphi }_{\lambda \mu }={\sigma }_{\lambda }$ whenever $\lambda \le \mu$, then $\sigma$ which is defined above $(\sigma :\overline{X}\to Y;\overline{x}\mapsto {\sigma }_{\lambda }\left(x\right)$ for any $x\in X_{\lambda })$ is a spectral morphism of quantum B-algebras.

Indeed, suppose that $y\in Y$ and $\overline{x}\in \overline{X}$, where $x\in X_{\lambda }$. Since ${\sigma }_{\lambda }:X_{\lambda }\to Y$ is a spectral morphism, we have $y{\to }_Y{\sigma }_{\lambda }\left(x\right)$ belongs to ${\sigma }_{\lambda }\left(X_{\lambda }\right)$, and so $y{\to }_Y{\sigma }_{\lambda }\left(x\right)={\sigma }_{\lambda }\left(d\right)$ for some $d\in X_{\lambda }$. It follows that

$$y{\to }_Y\sigma \left(\overline{x}\right)=y{\to }_Y{\sigma }_{\lambda }\left(x\right)={\sigma }_{\lambda }\left(d\right)=\sigma \left(\overline{d}\right)\in \sigma \left(\overline{X}\right),$$

and hence $\sigma$ is a spectral morphism of quantum B-algebras. □

The above theorem tells us that direct limits for any directed system in the category $qBE$ (or $qB$) exist. We will show that direct limits for any directed system in $qBU$ also exist.

Corollary 1.

Let $(X_{\lambda };{\to }_{\lambda },{⇝}_{\lambda },u_{\lambda },{\le }_{\lambda })$ be a unital quantum B-algebra for every $\lambda \in D$, where $u_{\lambda }$ is the unit element of $X_{\lambda }$. If ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ is a directed system in the category $qBU$, then direct limits for ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ exist in the category $qBU$.

Proof. 

Suppose that the assumption in the corollary is fulfilled. To prove that direct limits for directed system ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ exist in $qBU$, it suffices to show that the following statements (i), (ii) and (iii) hold.

(i)

The quantum B-algebra $\overline{X}$ defined in the proof of Theorem 1 is unital.

Indeed, for any $\lambda ,\mu \in D$, let $\nu \in D$ such that $\nu \ge \lambda ,\mu$. Since ${\varphi }_{\lambda \nu }:X_{\lambda }\to X_{\nu }$ and ${\varphi }_{\mu \nu }:X_{\mu }\to X_{\nu }$ are unital, we have ${\varphi }_{\lambda \nu }\left(u_{\lambda }\right)=u_{\nu }={\varphi }_{\mu \nu }\left(u_{\mu }\right)$, and so $\overline{u_{\lambda }}=\overline{u_{\mu }}$. Furthermore, for any $\overline{x}\in \overline{X}$, where $x\in X_{\alpha }$, let $\gamma \in D$ such that $\gamma \ge \lambda ,\alpha$, since ${\varphi }_{\alpha \gamma }\left(x\right)\equiv x$, we have

$$\overline{u_{\lambda }}\to \overline{x}=\overline{{\varphi }_{\lambda \gamma }\left(u_{\lambda }\right){\to }_{\gamma }{\varphi }_{\alpha \gamma }\left(x\right)}=\overline{u_{\gamma }{\to }_{\gamma }{\varphi }_{\alpha \gamma }\left(x\right)}=\overline{{\varphi }_{\alpha \gamma }\left(x\right)}=\overline{x}.$$

Similarly, we can obtain that $\overline{u_{\lambda }}⇝\overline{x}=\overline{x}$. Hence $\overline{u_{\lambda }}$ is the unit of $\overline{X}$.

(ii)

For each $\lambda \in D$, the ${\varphi }_{\lambda }$ defined in the proof of Theorem 1 is unital.

Indeed, we have shown in the proof of Theorem 1 that ${\varphi }_{\lambda }$ is an exact morphism of quantum B-algebra. Also, since ${\varphi }_{\lambda }\left(u_{\lambda }\right)=\overline{u_{\lambda }}$, we get that ${\varphi }_{\lambda }$ is unital.

(iii)

Let $Y=(Y,{\to }_Y,{⇝}_Y,u_Y{\le }_Y)$ be a unital quantum B-algebra. For each $\mu \in D$, if ${\sigma }_{\mu }:X_{\mu }\to Y$ is a unital morphism such that ${\sigma }_{\mu }{\varphi }_{\lambda \mu }={\sigma }_{\lambda }$ whenever $\lambda \le \mu$, then the function $\sigma$ defined in the proof of Theorem 1 ($\sigma :\overline{X}\to Y;\overline{x}\mapsto {\sigma }_{\lambda }\left(x\right)$ for any $x\in X_{\lambda }$) is unital.

In fact, it is shown that $\sigma$ is exact in the proof of Theorem 1. Also, since ${\sigma }_{\lambda }$ is unital, we have $\sigma \left(\overline{u_{\lambda }}\right)={\sigma }_{\lambda }\left(u_{\lambda }\right)=u_Y$, and hence $\sigma$ is unital. □

3. Inverse Limits of Quantum $\mathit{B}$-Algebras

The inverse limit notion is dual to that of direct limit. In this section, we will study inverse limits of quantum B-algebras. Recall that $\mathcal{C}$ means an arbitrary category.

Definition 3

([12]). An inverse system in $\mathcal{C}$ is a pair ${(A_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ satisfying the following conditions:

1. 

$(D,\le )$ is an upward directed set.

2. 

$A_{\lambda }\in ob\mathcal{C}$ for each $\lambda \in D$.

3. 

$A_{\lambda }\cap A_{\mu }=\varnothing$ if $\lambda \ne \mu$ in D.

4. 

If $\lambda ,\mu \in D$ with $\lambda \ge \mu$, then ${\varphi }_{\mu }^{\lambda }\in Ho{m}_{\mathcal{C}}(A_{\lambda },A_{\mu })$.

5. 

${\varphi }_{\nu }^{\mu }{\varphi }_{\mu }^{\lambda }={\varphi }_{\nu }^{\lambda }$ whenever $\lambda \ge \mu \ge \nu$ in D.

6. 

${\varphi }_{\lambda }^{\lambda }=i{d}_{A_{\lambda }}$, where $i{d}_{A_{\lambda }}$ is the identity map of $A_{\lambda }$ for all $\lambda \in D$.

Thus if $\lambda \ge \mu \ge \nu$ and $\lambda \ge \delta \ge \nu$ in D, then ${\varphi }_{\nu }^{\mu }{\varphi }_{\mu }^{\lambda }={\varphi }_{\nu }^{\lambda }={\varphi }_{\nu }^{\delta }{\varphi }_{\delta }^{\lambda }$.

Definition 4

([8]). An inverse limit for an inverse system ${(A_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ in $\mathcal{C}$ is a pair $(H;{\left\{{\psi }_{\lambda }\right\}}_{\lambda \in D})$, where $H\in ob\mathcal{C}$ and each ${\psi }_{\lambda }\in Ho{m}_{\mathcal{C}}(H,A_{\lambda })$, such that

1. 

${\varphi }_{\mu }^{\lambda }{\psi }_{\lambda }={\psi }_{\mu }$ whenever $\lambda \ge \mu$ in D.

2. 

For any $K\in ob\mathcal{C}$, if ${\pi }_{\lambda }\in Ho{m}_{\mathcal{C}}(K,A_{\lambda })$ such that ${\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }={\pi }_{\mu }$ for all $\lambda ,\mu \in D$ with $\lambda \ge \mu$, then there exists a unique $\pi \in Ho{m}_{\mathcal{C}}(K,H)$ such that ${\pi }_{\mu }={\psi }_{\mu }\pi$ for all $\mu \in D$.

By Definition 4, we know that if both $(H;{\left\{{\psi }_{\lambda }\right\}}_{\lambda \in D})$ and $(K;{\left\{{\pi }_{\lambda }\right\}}_{\lambda \in D})$ are inverse limits for an inverse system ${(A_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ in $\mathcal{C}$, then H is isomorphic to K.

Now, let ${(X_{\lambda };{\{{\varphi }_{\mu }^{\lambda }\}}_{\lambda \ge \mu })}_{\lambda \in D}$ be an inverse system in the category $qBE$. Consider the following subset of the direct product ${\prod }_{\lambda \in D}{X}_{\lambda }$:

$\widehat{H}=\{\widehat{a}\in {\prod }_{\lambda \in D}{X}_{\lambda }|{\varphi }_{\mu }^{\lambda }\left(a_{\lambda }\right)=a_{\mu }$ for all $\lambda \ge \mu \}$, where $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D}$.

When $\widehat{H}\ne \varnothing$, we define two operations ↠ and ↣ on $\widehat{H}$ as follows:

$$\widehat{a}\twoheadrightarrow \widehat{b}={\left(a_{\lambda }{\to }_{\lambda }{b}_{\lambda }\right)}_{\lambda \in D},\mathrm{and}\widehat{a}\rightarrowtail \widehat{b}={\left(a_{\lambda }{⇝}_{\lambda }{b}_{\lambda }\right)}_{\lambda \in D},$$

where $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D},\widehat{b}={\left(b_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$. We have $\widehat{a}\twoheadrightarrow \widehat{b}\in \widehat{H}$ and $\widehat{a}\rightarrowtail \widehat{b}\in \widehat{H}$. Indeed, let $\lambda ,\mu \in D$ such that $\lambda \ge \mu$. Noticing that ${\varphi }_{\mu }^{\lambda }$ is an exact morphism of quantum B-algebras, we get

$${\varphi }_{\mu }^{\lambda }\left(a_{\lambda }{\to }_{\lambda }{b}_{\lambda }\right)={\varphi }_{\mu }^{\lambda }\left(a_{\lambda }\right){\to }_{\mu }{\varphi }_{\mu }^{\lambda }\left(b_{\lambda }\right)=a_{\mu }{\to }_{\mu }{b}_{\mu }$$

and

$${\varphi }_{\mu }^{\lambda }\left(a_{\lambda }{⇝}_{\lambda }{b}_{\lambda }\right)={\varphi }_{\mu }^{\lambda }\left(a_{\lambda }\right){⇝}_{\mu }{\varphi }_{\mu }^{\lambda }\left(b_{\lambda }\right)=a_{\mu }{⇝}_{\mu }{b}_{\mu },$$

whence $\widehat{a}\twoheadrightarrow \widehat{b}\in \widehat{H}$ and $\widehat{a}\rightarrowtail \widehat{b}\in \widehat{H}$.

Moreover, define a binary relation ≦ on $\widehat{H}$ by: for all $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D},\widehat{b}={\left(b_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$,

$$\widehat{a}\leqq \widehat{b}⟺(\forall \lambda \in D)a_{\lambda }{\le }_{\lambda }{b}_{\lambda }.$$

Then it is easy to see that ≦ is a partial order on $\widehat{H}$.

Remark 1.

If the upward directed set D has a maximum element ϖ, then $\widehat{H}\ne \varnothing$. Indeed, for any $z_{\varpi }\in X_{\varpi }$, put $z_{\mu }={\varphi }_{\mu }^{\varpi }\left(z_{\varpi }\right)$ for any $\mu \in D$, and let $\widehat{z}={\left(z_{\mu }\right)}_{\mu \in D}$. Then we have $\widehat{z}\in \widehat{H}$, since ${\varphi }_{\mu }^{\lambda }\left(z_{\lambda }\right)={\varphi }_{\mu }^{\lambda }\left({\varphi }_{\lambda }^{\varpi }\left(z_{\varpi }\right)\right)={\varphi }_{\mu }^{\varpi }\left(z_{\varpi }\right)=z_{\mu }$ for all $\lambda \ge \mu$.

For each $\lambda \in D$, define a map ${\psi }_{\lambda }:\widehat{H}\to X_{\lambda }$ by ${\psi }_{\lambda }\left(\widehat{a}\right)=a_{\lambda }$, where $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$.

Next, we will show that if D is an upward directed set with a maximum element, then $(\widehat{H};\twoheadrightarrow ,\rightarrowtail ,\leqq )$ is a quantum B-algebra and $(\widehat{H},{\left\{{\psi }_{\lambda }\right\}}_{\lambda \in D})$ is an inverse limit of the inverse system ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ in the category $qBE$. For this purpose, the following lemma is needed.

Lemma 3.

Let ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ be an inverse system in the category $qBE$ (resp. $qB)$. If D has a maximum element ϖ, then the following statements are true:

(i) 

$(\widehat{H};\twoheadrightarrow ,\rightarrowtail ,\leqq )$ is a quantum B-algebra.

(ii) 

For each $\lambda \in D$, the map ${\psi }_{\lambda }\in Ho{m}_{qBE}(\widehat{H},X_{\lambda })$ (resp. ${\psi }_{\lambda }\in Ho{m}_{qB}(\widehat{H},X_{\lambda }))$, and ${\psi }_{\mu }={\varphi }_{\mu }^{\lambda }{\psi }_{\lambda }$ whenever $\lambda \ge \mu$ in D.

(iii) 

For any quantum B-algebra K, if for each $\lambda \in D$, ${\pi }_{\lambda }\in Ho{m}_{qBE}(K,X_{\lambda })$ (resp. ${\pi }_{\lambda }\in Ho{m}_{qB}(K,X_{\lambda }))$ such that ${\pi }_{\mu }={\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }$ whenever $\lambda \ge \mu$ in D, then there exists a unique $\pi \in Ho{m}_{qBE}(K,\widehat{H})($resp. $\pi \in Ho{m}_{qB}(K,\widehat{H}))$ such that ${\pi }_{\mu }={\psi }_{\mu }\pi$ for all $\mu \in D$.

Proof. 

Assume that D is an upward directed set with a maximum element $\varpi$, and ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ is an inverse system in $qBE$. Let $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D},\widehat{b}={\left(b_{\lambda }\right)}_{\lambda \in D}$, $\widehat{c}={\left(c_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$.

(i) If $\widehat{a}\leqq \widehat{b}$, then for any $\lambda \in D$, we have $a_{\lambda }{\le }_{\lambda }{b}_{\lambda }$, and so $c_{\lambda }{\to }_{\lambda }{a}_{\lambda }{\le }_{\lambda }{c}_{\lambda }{\to }_{\lambda }{b}_{\lambda }$ by Equation (3). Thus, $\widehat{c}\twoheadrightarrow \widehat{a}\leqq \widehat{c}\twoheadrightarrow \widehat{b}$ by the definitions of ↠ and ≦, and therefore $(\widehat{H};\twoheadrightarrow ,\rightarrowtail ,\leqq )$ satisfies Equation (3). Also, by the definitions of ≦, ↣, ↠, and Equations (4) and (5) we have

$$\begin{array}{ccc}& & \widehat{a}\leqq \widehat{b}\twoheadrightarrow \widehat{c}\hfill \\ & \iff & (\forall \lambda \in D)a_{\lambda }{\le }_{\lambda }{b}_{\lambda }\to c_{\lambda }\hfill \\ & \iff & (\forall \lambda \in D)b_{\lambda }{\le }_{\lambda }{a}_{\lambda }{⇝}_{\lambda }{c}_{\lambda }\hfill \\ & \iff & \widehat{b}\leqq \widehat{a}\rightarrowtail \widehat{c};\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \widehat{a}\twoheadrightarrow (\widehat{b}\rightarrowtail \widehat{c})\hfill \\ & =& {\left(a_{\lambda }{\to }_{\lambda }\left(b_{\lambda }{⇝}_{\lambda }{c}_{\lambda }\right)\right)}_{\lambda \in D}\hfill \\ & =& {\left(b_{\lambda }{⇝}_{\lambda }\left(a_{\lambda }{\to }_{\lambda }{c}_{\lambda }\right)\right)}_{\lambda \in D}\hfill \\ & =& \widehat{b}\rightarrowtail (\widehat{a}\twoheadrightarrow \widehat{c}).\hfill \end{array}$$

Thus, $(\widehat{H};\twoheadrightarrow ,\rightarrowtail ,\leqq )$ is a quantum B-algebra by Lemma 1.

(ii) For each $\lambda \in D$, by the definition of ${\psi }_{\lambda }$ we have

$${\psi }_{\lambda }(\widehat{a}\twoheadrightarrow \widehat{b})=a_{\lambda }{\to }_{\lambda }{b}_{\lambda }={\psi }_{\lambda }\left(\widehat{a}\right){\to }_{\lambda }{\psi }_{\lambda }\left(\widehat{b}\right),$$

$${\psi }_{\lambda }(\widehat{a}\rightarrowtail \widehat{b})=a_{\lambda }{⇝}_{\lambda }{b}_{\lambda }={\psi }_{\lambda }\left(\widehat{a}\right){⇝}_{\lambda }{\psi }_{\lambda }\left(\widehat{b}\right)$$

and

$$\widehat{a}\leqq \widehat{b}⟹a_{\lambda }{\le }_{\lambda }{b}_{\lambda }⟹{\psi }_{\lambda }\left(\widehat{a}\right){\le }_{\lambda }{\psi }_{\lambda }\left(\widehat{b}\right).$$

Thus, ${\psi }_{\lambda }\in Ho{m}_{qBE}(\widehat{H},X_{\lambda })$.

If $\lambda \ge \mu$, then ${\varphi }_{\mu }^{\lambda }{\psi }_{\lambda }\left(\widehat{a}\right)={\varphi }_{\mu }^{\lambda }\left(a_{\lambda }\right)=a_{\mu }={\psi }_{\mu }\left(\widehat{a}\right)$, and so ${\varphi }_{\mu }^{\lambda }{\psi }_{\lambda }={\psi }_{\mu }$.

Moreover, if ${\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu }$ is a family of spectral morphisms of quantum B-algebras, then ${\psi }_{\lambda }\in Ho{m}_{qB}(\widehat{H},X_{\lambda })$ for every $\lambda \in D$. In fact, let $d_{\lambda }\in X_{\lambda }$ and $h_{\lambda }\in {\psi }_{\lambda }\left(\widehat{H}\right)$. Then $h_{\lambda }={\psi }_{\lambda }\left(\widehat{h}\right)$ for some $\widehat{h}={\left(h_{\mu }\right)}_{\mu \in D}\in \widehat{H}$. Since $\varpi$ is the maximum element of D, we have ${\psi }_{\lambda }={\varphi }_{\lambda }^{\varpi }{\psi }_{\varpi }$, and so

$$h_{\lambda }={\psi }_{\lambda }\left(\widehat{h}\right)={\varphi }_{\lambda }^{\varpi }{\psi }_{\varpi }\left(\widehat{h}\right)={\varphi }_{\lambda }^{\varpi }\left(h_{\varpi }\right)\in {\varphi }_{\lambda }^{\varpi }\left(X_{\varpi }\right).$$

It follows that $d_{\lambda }{\to }_{\lambda }{h}_{\lambda }$ belongs to ${\varphi }_{\lambda }^{\varpi }\left(X_{\varpi }\right)$, since ${\varphi }_{\lambda }^{\varpi }:X_{\varpi }\to X_{\lambda }$ is a spectral morphism of quantum B-algebras. Thus there exists $g_{\varpi }\in X_{\varpi }$ such that

$$d_{\lambda }{\to }_{\lambda }{h}_{\lambda }={\varphi }_{\lambda }^{\varpi }\left(g_{\varpi }\right).$$

Let $g_{\mu }={\varphi }_{\mu }^{\varpi }\left(g_{\varpi }\right)$. Then $\widehat{g}={\left(g_{\mu }\right)}_{\mu \in D}\in \widehat{H}$ by Remark 1. Since ${\varphi }_{\lambda }^{\varpi }{\psi }_{\varpi }={\psi }_{\lambda }$, we immediately get

$$d_{\lambda }{\to }_{\lambda }{h}_{\lambda }={\varphi }_{\lambda }^{\varpi }\left(g_{\varpi }\right)={\varphi }_{\lambda }^{\varpi }{\psi }_{\varpi }\left(\widehat{g}\right)={\psi }_{\lambda }\left(\widehat{g}\right)\in {\psi }_{\lambda }\left(\widehat{H}\right),$$

an thus ${\psi }_{\lambda }$ is a spectral morphism, i.e., ${\psi }_{\lambda }\in Ho{m}_{qB}(\widehat{H},X_{\lambda })$.

(iii) Assume that $K=(K;{\to }_K,{⇝}_K,{\le }_K)$ is a quantum B-algebra. For each $\lambda \in D$, suppose that ${\pi }_{\lambda }\in Ho{m}_{qBE}(K,X_{\lambda })$ such that ${\pi }_{\mu }={\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }$ whenever $\lambda \ge \mu$.

Define a map $\pi :K\to \widehat{H}$ by $\pi \left(x\right)={\left({\pi }_{\lambda }\left(x\right)\right)}_{\lambda \in D}$, where $x\in K$. Since ${\pi }_{\mu }={\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }$ whenever $\lambda \ge \mu$, we have ${\varphi }_{\mu }^{\lambda }\left({\pi }_{\lambda }\left(x\right)\right)={\pi }_{\mu }\left(x\right)$, which implies that ${\left({\pi }_{\lambda }\left(x\right)\right)}_{\lambda \in D}\in \widehat{H}$, and hence $\pi$ is well defined. Also, for any $x\in K$ and $\mu \in D$, we have ${\psi }_{\mu }\pi \left(x\right)={\psi }_{\mu }\left({\left({\pi }_{\lambda }\left(x\right)\right)}_{\lambda \in D}\right)={\pi }_{\mu }\left(x\right)$, and so ${\psi }_{\mu }\pi ={\pi }_{\mu }$.

Next, we will show that $\pi \in Ho{m}_{qBE}(K,\widehat{H})$. In fact, for all $x,y\in K$, since ${\pi }_{\lambda }\in Ho{m}_{qBE}(K,X_{\lambda })$ for each $\lambda \in D$, we have by the definitions ↠ and ↣ on $\widehat{H}$ that

$$\pi \left(x{\to }_{K}y\right)={\left({\pi }_{\lambda }\left(x{\to }_{K}y\right)\right)}_{\lambda \in D}={\left({\pi }_{\lambda }\left(x\right){\to }_{\lambda }{\pi }_{\lambda }\left(y\right)\right)}_{\lambda \in D}=\pi \left(x\right)\twoheadrightarrow \pi \left(y\right);$$

$$\pi \left(x{⇝}_{K}y\right)={\left({\pi }_{\lambda }\left(x{⇝}_{K}y\right)\right)}_{\lambda \in D}={\left({\pi }_{\lambda }\left(x\right){⇝}_{\lambda }{\pi }_{\lambda }\left(y\right)\right)}_{\lambda \in D}=\pi \left(x\right)\rightarrowtail \pi \left(y\right);$$

and if $x{\le }_{K}y$, then ${\pi }_{\lambda }\left(x\right){\le }_{\lambda }{\pi }_{\lambda }\left(y\right)$, so

$$\pi \left(x\right)={\left({\pi }_{\lambda }\left(x\right)\right)}_{\lambda \in D}\leqq {\left({\pi }_{\lambda }\left(y\right)\right)}_{\lambda \in D}=\pi \left(y\right).$$

Therefore, we have shown that $\pi \in Ho{m}_{qBE}(K,\widehat{H})$.

Moreover, if ${\pi }_{\lambda }\in Ho{m}_{qB}(K,X_{\lambda })$ such that ${\pi }_{\mu }={\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }$ whenever $\lambda \ge \mu$, then $\pi \in Ho{m}_{qB}(K,\widehat{H})$. Indeed, let $\widehat{p}={\left(p_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$ and $\widehat{q}\in \pi \left(K\right)$. Then $\widehat{q}=\pi \left(z\right)={\left({\pi }_{\lambda }\left(z\right)\right)}_{\lambda \in D}$ for some $z\in K$. Since ${\pi }_{\lambda }$ is a spectral morphism, we have $p_{\lambda }{\to }_{\lambda }{\pi }_{\lambda }\left(z\right)$ belongs to ${\pi }_{\lambda }\left(K\right)$, and so $p_{\lambda }{\to }_{\lambda }{\pi }_{\lambda }\left(z\right)={\pi }_{\lambda }\left(v\right)$ for some $v\in K$. It follows that

$$\widehat{p}\twoheadrightarrow \widehat{q}=\widehat{p}\twoheadrightarrow \pi \left(z\right)={\left(p_{\lambda }{\to }_{\lambda }{\pi }_{\lambda }\left(z\right)\right)}_{\lambda \in D}={\left({\pi }_{\lambda }\left(\nu \right)\right)}_{\lambda \in D}=\pi \left(v\right)\in \pi \left(K\right),$$

and thus $\pi \in Ho{m}_{qB}(K,\widehat{H})$.

Finally, to show that such $\pi$ is unique, let $\tau \in Ho{m}_{qBE}(K,\widehat{H})$ such that ${\pi }_{\mu }={\psi }_{\mu }\tau$ for all $\mu \in D$. Then ${\psi }_{\mu }\left(\tau \left(x\right)\right)={\pi }_{\mu }\left(x\right)$ for any $x\in K$. Recall that ${\psi }_{\mu }:\widehat{H}\to X_{\mu }$ is defined by ${\psi }_{\mu }\left(\widehat{a}\right)=a_{\mu }$, where $\widehat{a}={\left(a_{\mu }\right)}_{\mu \in D}\in \widehat{H}$. We get $\tau \left(x\right)={\left({\pi }_{\mu }\left(x\right)\right)}_{\mu \in D}=\pi \left(x\right)$, and thus $\tau =\pi$. □

Theorem 2.

Let ${(X_{\lambda };{\{{\varphi }_{\mu }^{\lambda }\}}_{\lambda \ge \mu })}_{\lambda \in D}$ be an inverse system in the category $qBE$ (resp. $qB)$. If D has a maximum element ϖ, then $(\widehat{H},{\left\{{\psi }_{\lambda }\right\}}_{\lambda \in D})$ is an inverse limit of ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ in the category $qBE($resp. $qB)$.

Moreover, if $X_{\lambda }$ is commutative for each $\lambda \in D$, then inverse limits for ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ are also commutative.

Proof. 

The first part follows immediately from Lemma 3 and Definition 4.

If $X_{\lambda }$ is commutative for each $\lambda \in D$, then it is easy to verify that $(\widehat{H};\twoheadrightarrow ,\rightarrowtail ,\leqq )$ is a commutative quantum B-algebra, and hence inverse limits for ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ are also commutative. □

Proposition 2.

Let $(X_{\lambda };{\to }_{\lambda },{⇝}_{\lambda },u_{\lambda },{\le }_{\lambda })$ be a unital quantum B-algebra for each $\lambda \in D$, and ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ be an inverse system in the category $qBU$. If D has a maximum element ϖ, then inverse limits for ${(X_{\lambda };{\left\{{\varphi }_{\lambda \mu }\right\}}_{\lambda \le \mu })}_{\lambda \in D}$ exist in the category $qBU$.

Proof. 

Suppose that the assumption in the proposition is fulfilled. To prove that inverse limits for the inverse system ${(X_{\lambda };{\left\{{\varphi }_{\mu }^{\lambda }\right\}}_{\lambda \ge \mu })}_{\lambda \in D}$ exist in the category $qBU$, it suffices to show that:

(i) The quantum B-algebra $\widehat{H}$ is unital.

Indeed, let $\widehat{u}={\left(u_{\lambda }\right)}_{\lambda \in D}$, where $u_{\lambda }$ is the unital element of $X_{\lambda }$. For all $\lambda \ge \mu$, since ${\varphi }_{\mu }^{\lambda }$ is unital, we have ${\varphi }_{\mu }^{\lambda }\left(u_{\lambda }\right)=u_{\mu }$, and so $\widehat{u}\in \widehat{H}$. Also, for any $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$, we have

$$\widehat{u}\twoheadrightarrow \widehat{a}={\left(u_{\lambda }{\to }_{\lambda }{a}_{\lambda }\right)}_{\lambda \in D}={\left(a_{\lambda }\right)}_{\lambda \in D}=\widehat{a}$$

and

$$\widehat{u}\rightarrowtail \widehat{a}={\left(u_{\lambda }{⇝}_{\lambda }{a}_{\lambda }\right)}_{\lambda \in D}={\left(a_{\lambda }\right)}_{\lambda \in D}=\widehat{a}.$$

Hence $\widehat{u}$ is the unit element of $\widehat{H}$.

(ii) For each $\lambda \in D$, ${\psi }_{\lambda }$ is unital, where ${\psi }_{\lambda }:\widehat{H}\to X_{\lambda }$ is defined by ${\psi }_{\lambda }\left(\widehat{a}\right)=a_{\lambda }$ for any $\widehat{a}={\left(a_{\lambda }\right)}_{\lambda \in D}\in \widehat{H}$.

Indeed, we know by Lemma 3 (iii) that ${\psi }_{\lambda }$ is an exact morphism of quantum B-algebras. Also, since ${\psi }_{\lambda }\left(\widehat{u}\right)=u_{\lambda }$, we get that ${\psi }_{\lambda }$ is unital.

(iii) Let $K=(K;{\to }_K,{⇝}_K,u_K{\le }_K)$ be a unital quantum B-algebra. If for each $\lambda \in D$, ${\pi }_{\lambda }\in Ho{m}_{qBU}(K,X_{\lambda })$ such that ${\pi }_{\mu }={\varphi }_{\mu }^{\lambda }{\pi }_{\lambda }$ whenever $\lambda \ge \mu$, then $\pi$ (recall that $\pi :K\to \widehat{H}$ is defined by $\pi \left(x\right)={\left({\pi }_{\lambda }\left(x\right)\right)}_{\lambda \in D}$ for any $x\in K$) is unital.

In fact, it is shown that $\pi$ is exact in the proof of Lemma 3. Also, since ${\pi }_{\lambda }$ is unital, we have $\pi \left(u_K\right)={\left({\pi }_{\lambda }\left(u_K\right)\right)}_{\lambda \in D}={\left(u_{\lambda }\right)}_{\lambda \in D}=\widehat{u}$, and hence $\pi$ is unital. □

4. Conclusions

In this paper, we guarantee the existences of both direct limits and inverse limits in the categories of quantum B-algebras with morphisms of exact ones or spectral ones. We will further study category theory of quantum B-algebras, and try to find some applications.