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\fancyhead[CE]{J. Kazemiasl, F. Khaksar Haghani and Sh. Heidarian}
\fancyhead[CO]{Some Relations on Noetherian and Boolean Artinian $BL$-algebras}



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{\noindent Journal of Mathematical Extension} \\
%Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
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{\Large \bf
Some Relations on Noetherian and Boolean Artinian $BL$-algebras\\}
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\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Jamal Kazemiasl}$~^1$\vspace*{2mm}\\
%\vspace{2mm} {\small  Department of Mathematics, Faculty of Basic Science, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.} \vspace{2mm}

{\bf  Farhad Khaksar Haghani$~^{2,*}$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{2mm}\\
%\vspace{2mm} {\small   Department of Mathematics, Faculty of Basic Science, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.} \vspace{2mm}\\
{\bf  Shahram Heidadian$~^3$ \vspace*{2mm}
\let\thefootnote\relax\footnote{}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.}$~^{1,2,3}$ \vspace{2mm}
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{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, we derive some new result on Noetherian and Boolean Artinian $BL$-algebras. We further obtain some relations between local and semilocal $BL$-algebras and Boolean Artinian $BL$-algebras.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification 2010:} 06D99; 08A99; 03G99.

\noindent{\bf Keywords and Phrases:} Noetherian $BL$-algebra; Boolean Artinian $BL$-algebra; Prime filters; Generated filters; Deductive system.
\end{quotation}}

\section{Introduction}
	
	$BL$-algebras have been invented by H\'ajek [2] in order to provide an algebraic proof of the completeness theorem of "basic logic" ($BL$, for short) arising from the continuous triangular norms, familiar in the fuzzy logic frame-work. $BL$-algebras are the algebraic structures for H\'ajek [2] basic logic in order to investigate many-valued logic by algebraic means. He provided an algebraic counterpart of a propositional logic, ($BL$), which embodies a fragment common to some of the most important many-valued logics, namely Lukasiewicz Logic, G\"{o}del Logic and Product Logic. This Basic Logic
	 is proposed as “the most general” many-valued logic with truth values in interval [0, 1]
	and $BL$-algebras are the corresponding Lindenbaum–Tarski algebras. The language of propositional H\'ajek basic logic [2] contains the binary connectives $o$, $\Rightarrow $ and the constant $ \overline{0}$. 	Axioms of $BL$ are given as:
	
	(A1) ($\varphi \Rightarrow \psi)\Rightarrow ((\psi \Rightarrow w)\Rightarrow (\varphi \Rightarrow w)); $
	
	(A2) ($\varphi o\psi)\Rightarrow \varphi; $
	
	(A3) ($\varphi o\psi)\Rightarrow \left(\psi o\varphi \right); $
	
	(A4) ($\varphi o\left(\varphi \Rightarrow \psi \right))\Rightarrow \left(\psi o\left(\psi \Rightarrow \varphi \right)\right)$;
	
	(A5a) ($\varphi \Rightarrow (\psi \Rightarrow w))\Rightarrow ((\varphi o\psi )\Rightarrow w); $
	
	(A5b) (($\varphi o\psi )\Rightarrow w)\Rightarrow (\varphi \Rightarrow (\psi \Rightarrow w));$
	
	(A6) (($\varphi \Rightarrow \psi )\Rightarrow w)\Rightarrow ((\psi \Rightarrow \varphi )\Rightarrow w)\Rightarrow w); $
	
	(A7) $\overline{0 }\Rightarrow w. $\\
	
	H\'ajek [2] introduced the concepts of filters and prime filters in $BL$-algebras. From logical point of view, filters correspond to sets of provable formula. E. Turunen studied some properties of filters theory, which plays important role in studying logical algebras. He showed how $BL$-algebras can be studied by deductive systems. Deductive systems correspond to subsets closed with respect to Modus Ponens and they are called filters, too. Also he introduce the notion of Boolean Artinian $BL$-algebras [12]. Motamed and Moghaderi [5], introduced the notions of Noetherian and Artinian on $BL$-algebras. They obtained some equivalent definitions of Noetherian and Artinian  $BL$-algebras. Meng B. L and Xin X. L [6], introduced the co-Noehterian $BL$-algebras. \\In this paper, we obtain some new relations between the above notions.
	The structure of the paper is as follows: in section 2, we recall some definitions and results about $BL$-algebras that we use in the sequel. In section 3, we recall the notion of Noetherian and Boolean Artinian $BL$-algebras and we derive some results about the relations between them.
	
		\section{Preliminaries}
			
	In this section, we give some definitions and theorems which are needed in the rest of the paper.
	
	An algebra$(A, \wedge, \vee, { \odot }, \longrightarrow, 0, 1)$ of the type (2, 2, 2, 2, 0, 0) is called a $BL$-algebra if satisfies the following axioms [2]:
	
	
	
	($BL$1) $(A, \wedge, \vee, 0, 1)$ is a bounded lattice;
	
	($BL$2) $(A, {\odot}, 1)$ is a commutative monoid;
	
	($BL$3) $ \odot$ and $\longrightarrow$ form an adjoint pair, i.e., $ c\leq a\longrightarrow b $ if and only if $ a\odot c\le b $;
	
	
	($BL$4) $a\wedge b=a{ \odot } \left(a\longrightarrow b\right)$;
	
	($BL$5) $(a\longrightarrow b)\vee \left(b\longrightarrow a\right)=1$.
	
	For all $a, b, c\in A$. We denote $\overline{x}=x\longrightarrow 0$ and $x^{--} =(\overline{x})^{-}, $ for all $x\in A$. \\Most familiar example of a $BL$-algebra is the unit interval $[0, 1]$ endowed with the structure induced by a continuous t-norm. In the rest of this paper $A$, denotes the universe of a $BL$-algebra [2].
	
	A $BL$-algebra is nontrivial if $0\ne 1$. For any $BL$-algebra $A$, the deduct $L(A)=(A, \wedge, \vee, 0, 1)$ is a bounded distributive lattice. We denote the set of all natural numbers by $\mathbb{N}$ and define $a^0=1$ and $a^{n} = a^{n-1}\odot a$, for $n\in\mathbb{N}\backslash \left\{0\right\}$.
	H\'ajek [2] defined a filter of a $BL$-algebra $A$ to be a nonempty subset $F$ of $A$ such that $(i)$ if a, b$ \in
	F$ implies $ a{ \odot }b\in F$, $(ii)$ if $ a\in F, a\le b$ then $ b\in F$. E. Turunen [7] defined a deductive system of a $BL$-algebra $A$ to be a nonempty subset $D$ of $A$ such that $(i)$ $1\in D$, $(ii)$ $x\in D$ and $x\longrightarrow y\in D$ imply $y\in D$. The set of all deductive systems of a $BL$-algebra $A$ is denoted by $\texttt{D}(A)$. A subset $F$ of a $BL$-algebra $A$ is a deductive system of $A$ if and only if $F$ is a filter of $A$ [7]. A deductive system $D$ of $BL-$algebra $A$ is Boolean if, for all $x\in A$, $x\vee \overline{x}\in D$ [9].
	
	A filter $F$ of a $BL$-algebra $A$ is proper if $F\ne A$. A proper filter $P$ of $A$ is called a prime filter of $A$ if for all $x, y\in $A$, x\vee y\in P$ implies $x\in P$ or $y\in P$. A proper filter $P$ of $A$ is Prime if and only if $P$ can not be expressed as an intersection of two filters properly containing $P$ or equivalently, for all $x, y\in $A$, $ either $ x\longrightarrow y\in P$ or $y\longrightarrow x \in P$ [8].
	
	If $F$, $G$ and $P$ are filters of $A$, then $P$ is a prime filter of $A$ if and only if $F\cap G\subseteq P$ then $F\subseteq P$ or $G\subseteq P$. \\
		A proper filter $M$ of $A$ is a maximal filter if and only if for any $x\notin M$ there exists $n\in \mathbb{N}$ such that ($x^{n} )^{-} \in M$ [7]. Every maximal filter of $A$ is a prime filter [8].
	
	The set of all filters, prime filters and maximal filters of a $BL$-algebra $A$ are denoted by $\digamma(A)$, $Spec(A)$ and $Max(A)$, respectively. For every subset $X\subseteq A$, the smallest filter of $A$ which
contains $X$, i.e., the intersection of all filters $F \in \digamma(A)$ such that $F \supseteq X$, is said to be the filter generated
by $X$ [1]. The filter of $A$ generated by $X$ will be denoted by $\left\langle X\right\rangle $, where $X\subseteq A$, in which $\left\langle \emptyset \right\rangle $=$\left\{1\right\}$ and $\left\langle X\right\rangle =\{a\in A:x_1\odot x_2\odot{ \dots \odot }x_n\leq a, $ for some $n\in\mathbb{N}$ and $x_1, x_2{, \dots, x}_n\in X\}$ [8]. A filter $F\in \digamma(A)$ is called finitely generated, if $F=\left\langle x_{1}, x_{2}, \ldots ,x_{n} \right\rangle$, for some $ x_{1, } \ldots, x_{n} \in $A$ $ and $n\in \mathbb{N}$. For $F\in \digamma(A)$ and $x\in A\backslash F$, $F\left\langle x\right\rangle = \left\langle F\bigcup \{ x\} \right\rangle$ and so $F\langle x\rangle =\{a\in A: a\ge f { \odot }x^{n}$, for some $f\in F$, and $n\ge 1 \}$ [8]. \\
	
	\noindent\textbf{Definition 2.1}([5]) Let $A$ be a $BL$-algebra and $F$ be a filter of $A$. A prime filter $P$ of $A$, is called a minimal prime filter of $F$ if ${ F}\subseteq { P}$ and if further, there exists $Q\in Spec\left({ A}\right)$ such that ${F}\subseteq { Q}\subseteq {P}$, then $P=Q$. The set of all minimal prime filter of $\left\{{ 1}\right\}$, is denoted by $Min(A)$. \\
	
	\noindent\textbf{Definition 2.2} ([5]) A $BL$-algebra $A$ is called Noetherian (Artinian), if for every increasing (decreasing) chain of its filters
	${ F}_{{ 1}} \subseteq { F}_{{ 2}} \subseteq \dots $\\$\left({ F}_{{ 1}} \supseteq { F}_{{ 2}} \supseteq \dots \right)$, there exists $n\in \mathbb{N}$ such that $ { F}_{{ i}} { =F}_{{ n}}$, for all ${ i}\ge { n}$. \\
	
		
	\noindent\textbf{Theorem 2.3} ([5]) Let ${A}$ be a $BL$-algebra. Then ${A}$ is Noetherian $BL$-algebra if and only if every filter of $A$ is finitely generated. \\
	
	\noindent\textbf{Definition 2.4}([2]) Let $A$ be a $BL$-algebra. A nonempty subset $I\subseteq A $ is called an ideal of $A$, if the following conditions are satisfied:
	
	$(i)$ $ 0\in I $,
	
	$(ii)$ If $x, (x^-\longrightarrow y^-)^-\in I $ then $ y\in I $. \\
	
	\noindent\textbf{Definition 2.5}([6]) A $BL$-algebra $A$ is said to be co-Noetherian with respect to ideals if every ideal of $A$ is finitely generated. A $BL$-algebra $A$ is satisfies the ascending chain condition with respect to its ideals if for every ascending chain sequence $ I_1\subseteq I_2\subseteq … $ of ideals of $A$, there exists $n\in \mathbb{N}$ such that $ I_i=I_n $, for all $ i\geq n$. \\
	
		
	\noindent\textbf{Remark 2.6} ([8]) Let $F$ and $G$ be two filters of $A$ such that $F \subseteq G$. It is evident that $\displaystyle\frac{G}{F }$ is a filter of $\displaystyle\frac{A}{F} $. Since $G$ is a filter, then it can be easily shown that $\displaystyle\frac{a}{F} \in \displaystyle\frac{G}{F} $ if and only if a $\in G$. Moreover, $\digamma(\displaystyle\frac{A}{F} )=\{ \displaystyle\frac{H}{F} : H\in \digamma(A), F \subseteq H\}$. \\


  \textbf{Definition 2.7} ([7]) A $BL$-algebra $A$ is called locally finite if all non-unit elements are of finite order, i.e., for any non-unit element $x$ of $A$, $x^n=0$ for some $n\ge 1$. Obviously, the only proper deductive system of a locally finite $BL$-algebra is $\{1\}$, thus, $M(A)= \{1\}$, where $M(A)=\bigcap \{M:M$ is a maximal deductive system of $A\}$. \\\\

  From [7], a $BL$-algebra $A$ is semisimple if $M(A)= \{1\}$. In [7], it is also proved that locally finite $BL$-algebras and locally finite $MV$-algebras are coincide. Moreover, for any $BL$-algebras $A$, $M$ is a maximal deductive system of $A$ if and only if $\displaystyle\frac{A}{M}$ is a locally finite $BL$-algebra if and only if for any $x\notin M$, $(x^n)^-\in M$ for some $n\ge 1$. E. Turunen [9], defined a $BL$-algebra $A$ is local if it has a unique maximal deductive system. He proved that a $BL$-algebra $A$ is local if and only if the unique maximal deductive system of $A$ is equal to $\{x\in A: x^n>0$ for all$\ n\ge 1\}$.
From [11], Given two locally finite $BL$-algebras $A$ and $B$, a product $BL$-algebra $A\times B$ contains two disjoint descending chains of deductive systems, namely $ A\times B \supseteq A\times \{1\}\supseteq \{(1, 1)\} $ and $A\times B \supseteq \{1\}\times B\supseteq \{(1, 1)\}$. Clearly, $A\times B$ is a semisimple $BL$-algebra and both maximal deductive system $A\times \{1\}$ and $\{1\}\times B$ are disjoint. More generally, given $n$ locally finite $BL$-algebras $A_1, A_2, A_3, \ldots A_n$, a product $BL$-algebra $\prod_{k=1}^n A_k$ is semisimple, contains $2^n-1$ proper deductive system and $n$ disjoint maximal deductive system $M_k=A_1\times\dots\times \{1\}\times\dots \times A_n$, $k= 1, 2, \ldots, n$. Also, any properly descending chain of deductive systems is finite. \\

\textbf{Proposition 2.8} ([11]) Let $A$ be a $BL$-algebra and $M_1, M_2, M_3, \ldots, M_n$ be $n$ maximal deductive systems, (not necessarily disjoint), of $A$. Then the product $BL$-algebra $\mathcal{A} = \prod_{k=1}^n A/M_k$ is semisimple, contains $2^n-1$ proper deductive systems, and $n$ disjoint maximal deductive systems further any properly descending chain of deductive system of $\mathcal{A}$ is finite. \\

\textbf{Definition 2.9} ([11]) A $BL$-algebra $A$ is semilocal if it contains only finite many disjoint maximal deductive systems. \\


\textbf{Definition 2.10} ([11]) A $BL$-algebra $A$ is Boolean Artinian if, any properly descending chain of Boolean deductive systems is finite. \\

\textbf{Definition 2.11} ([6]) Let $X$ be a subset of a $BL$-algebra $A$. The least ideal containing $X$ in $A$ is called the ideal generated by $X$ and denoted by $(X]$. If $X=\{a_1, a_2, \ldots, a_n\}$ then $(X]$ is denoted by $(a_1, a_2, \ldots, a_n]$ instead of $(\{a_1, a_2, \ldots, a_n\}]$. An ideal $I$ of $A$ is said to be finitely generated if there exist $a_1, a_2, \ldots, a_n\in A$ such that $I=(a_1, a_2, \ldots, a_n]$.\\

\textbf{Proposition 2.12} ([6]) Let $A$ be a $BL$-algebra. Then for any $x_1, \ldots, x_n\in A$ and $n\in \mathbb{N}$, $((x_1 ]\cup (x_2]\cup \dots \cup (x_n ]]=(( \bar{x}_1\odot \bar{x}_2\odot \dots \odot \bar{x}_n)^- ]$. \\
	
	\textbf{Definition 2.13} ([12]) Let $A$ be a $BL$-algebra. If $I$ is an ideals of $A$, $I= (x]$ where $x \in A$, then $I$ is called a principal ideal of $A$. \\
	
	
\noindent\textbf{Definition 2.14} ([8]) Let $A$ and $B$ be two $BL$-algebras. A map $f:A\longrightarrow B$ defined on $A$, is called a $BL$-homomorphism if, for all $x, y\in A, f\left(x\longrightarrow y\right)=f\left(x\right)\longrightarrow f\left(y\right)$,  $f\left(x{ \odot }y\right)=f\left(x\right){ \odot }f\left(y\right)$ and $f\left(0_{A} \right)=0_{B}$. Also, we define $ker(f)=\{a\in A: f(a)=1\}$ and $Im(f)=\{f(a):a\in A\}$. \\
	
	\noindent \textbf{Theorem 2.15 ([4])} Let $X$ be a subset of $A$. Then $\left\langle X\right\rangle =\{a\in A: (x_1\odot x_2\odot \dots \odot x_n)\longrightarrow a = 1$, for some $n\in \mathbb{ N}$ and $ x_1, x_2, \dots, \ x_n\in X\}$, where $\left\langle X\right\rangle$ is the filter of $A$ generated by $X$. \\

\noindent \textbf{Theorem 2.16 ([8])} A subset $F$ of a $BL$-algebra $A$ is a deductive system of $A$ if and only if $F$ is a filter of $A$. \\

\noindent \textbf{Theorem 2.17 ([3]) }Let $A$ be a nontrivial $BL$-algebra, $X\subseteq A$ and $D(X)=\{P\in Spec(A): X\nsubseteq P\}$. Then the following hold:

\noindent $(i)$ $X\subseteq Y\subseteq A$ implies $D(X)\subseteq D(Y)\subseteq  Spec(A)$;

\noindent $(ii)$ $D(\{0\})=Spec(A)$ and $D(\emptyset)=E(\{1\}) =\emptyset $;

\noindent $(iii)$ $D(X)=Spec(A)$ if and only if $ A =\left\langle  X \right\rangle $;

\noindent $(iv)$ $D(X) =\emptyset $ if and only if $ X=\emptyset $ or $ X=\{ 1\}$;

\noindent $(v)$  If $\{X_i\}_{i\in I}$ is any family of subsets of $ A$, then $ D (\bigcup_{ i\in  I}{X_i}) =\bigcap_{ i\in  I} D( X_{ i})$;

\noindent $(vi)$ $D(X)=D\left\langle  X \right\rangle$;

\noindent $(vii)$ $D(X)\bigcup  D(Y) = D(\left\langle  X\right\rangle \bigcup \left\langle  Y\right\rangle)$;

\noindent $(viii)$ If $ X, Y\subseteq  A$, then $\langle X\rangle=\langle Y\rangle$ if and only if $D(X)=D(Y)$;

\noindent $(ix)$  If $F$, $H$ are filters of $A$, then $ F=H$ if and only if $D(F)=D(H)$.\\

\noindent \textbf{Theorem 2.18 ([11])} Let $M_1,M_2,M_3,\dots,M_n$ be $n$ maximal deductive systems of a $BL$-algebra $A$ and ${\mathcal A}=\prod^n_{i=1}{{A}/{M_i}}$ be the product $BL$-algebra. Then a map $h:A\longrightarrow{\mathcal A}$ defined, for all $a\in A$, by
$h(a)=(a/M_1,a/{M_2},\dots ,{a}/{M_n})$
is an onto $BL$-homomorphism such that ${h}(a)= 1_{\mathcal A}$ if and only if ${a}\equiv 1$ mod $M_{i}$, for all $i=1,2,\dots ,n$ where, for $x,y\in A$, $x\equiv y$ mod $D$ iff $(x\longrightarrow y)\odot (y\longrightarrow x)\in D$. Thus ${A}/\bigcap^n_{i=1}{M_i}$ is isomorphic to ${\mathcal A} = \prod^n_{i=1}{{A}/{M_i}}$.

	
	\section{Some results on Noetherian and Boolean Artinian $BL$-algebras}
	
	
	In this section, we derive some new results on Noetherian, semilocal, local and Boolean Artinian $BL$-algebras.\\
	
	
	\textbf{Lemma 3.1} Let $A$ be a $BL$-algebra and $F\in \digamma(A)$. If $F$ is generated by a finite set of generators, then $F$ is generated by an element.
	
	\textbf{Proof.} If $F$ is generated by $\{x_1, x_2, \ldots, x_n\}$, for some $x_1, x_2, \ldots, x_n\in A$. Then $F$ is the smallest filter containing, $x_1, x_2, \ldots, x_n$. We claim	that $F$ is generated by $x_1\odot x_2\odot \dots \odot {{ x}}_{{ n}}$. It is enough to show that $F$ is the smallest filter of $A$, which contains ${{ x}}_{{ 1}}$$\odot {{ x}}_{{ 2}}\odot \dots \odot {{ x}}_{{ n}}$. Assume on the contrary that $F$ is not the smallest filter of $A$ with the above condition. Then, there exists a filter $G$ of $A$ which contains $x_1\odot x_2\odot\dots \odot x_n$ and $G\subset F$, so ${{ \ x}}_{{ 1}}\odot {{ (x}}_{{ 2}}\odot \dots \odot {{ x}}_{{ n}}{ \ )}\in  G$, then ${{ \ x}}_{{ 1}}\in G$ and $x_2\odot (x_3\odot \dots \odot x_n)\in G$. Therefore, ${{ \ x}}_{{ 1}}\in $ G, ${ \ }{{ \ x}}_{{ 2}}\in G$ and ${ \ }{{ x}}_{{ 3}}\odot \dots \odot {{ x}}_{{ n}}\in G$. Continuing this procedure, we conclude ${{ x}}_{{ 1}}, {{ x}}_{{ 2}}{, \dots, }{{ x}}_{{ n}}$ $\in G$, which is a contradiction. Thus, $F$ is the smallest filter of $A$ such that containing all ${{ x}}_{{ i}}$ for all 1$\le { i}\le { n}$. \\
				
	
	\noindent\textbf{Theorem 3.2} Let $A$ be a $BL$-algebra and A$\ne \emptyset$. Then $Spec(A)$ has minimal element with respect to inclusion.
	
	\textbf{Proof.}  Since $A\ne \emptyset $, so $Spec(A)\ne \emptyset$. We consider $(Spec(A), \supseteq { )}$ and Suppose $\left\{{ P}_{{ i}} \right\}$ be a chain in $(Spec(A), \supseteq { )}$. Put $P=\bigcap _{{ i}} { P}_{{ i}}$. Clearly $P$ is a filter of $A$. We prove $P$ is prime. Let $a\vee { b}\in { P} $ and ${ a}\notin { P}, { b}\notin { P}$. Then there exist $i, j\in \mathbb{N}, { a}\notin {P}_{{ i}}$, $b\notin { P}_{{ j}}$. Since $\left\{{ P}_{{ i}} \right\}$ is a chain, so ${ P}_{{ i}}  \supseteq { P}_{{ j}, }$  and ${ a}\notin { P}_{{ j}}. $ Therefore, $a\vee { b}\notin { P}_{{ j} }$ and ${ a}\vee { b}\notin { P}=\bigcap _{{ i}} { P}_{{ i}}$, which is a contradiction, so $P$ is prime and $Spec(A)$ has a minimal element with respect to inclusion. \\
		
	\noindent\textbf{Corollary 3.3} Let $A$ be a $BL$-algebra. If $A$ satisfy the ascending chain condition on finitely generated filters, then $A$ is a Noetherian $BL$-algebra. \\
	
	\textbf{Proof.} If $A$ has no nontrivial filter, it is clear that $A$ is a Noetherian $BL$-algebra. Suppose $A$ has at least one nontrivial filter, then the filter generated by an element of it, is a finitely generated filter and $A$ has a finitely generated filter. Put $F=\{ F_i \subseteq A: F_i$ is finitely generated filter of $A\}$.
	We prove that $F$ has a maximal element. Clearly $\left\langle1\right\rangle\in F$, then $F\ne \emptyset $. We show that $F$ has a maximal element. Let $F_{{ 1}} \in { F}$, if $F_{{ 1}}$ is maximal of $F$, it is obvious, otherwise, there exists $F_2 \in { F}$, $F_{{ 1}} \ne F_{{ 2}}$ such that ${ F}_{{ 1}} \subset $ ${ F}_{{ 2}}$. Now if ${F}_{{ 2}}$ is maximal element of $F$, we are done. But if $F_{{ 2}}$ is not maximal element of $F$, then there exist $ F_{{ 3}} \in { F}, ~ F_{{ 3}} \ne F_{{ 2}}$ and ${ F}_{{ 1}} \subset { F}_{{ 2}} \subset { F}_{{ 3}} $. Continuing this procedure, we have ${ F}_{{ 1}} \subset { F}_{{ 2}} \subset { F}_{{ 3}} \subset \dots \subset { F}_{{ n}} \subset \dots$. Since ${ F}_{{ i}} \in { F}$, for all $i\in \mathbb{ N}$, so every ${ F}_{{ i}}$ is finitely generated filter therefore, there exists $n\in \mathbb{ N}$ such that ${ F}_{{ i}} { =F}_{{ n}}$, for all ${ i}\ge { n}$. Hence ${F}_{{ n}}$ is a maximal element of $F$. To prove that $A$ is Noetherian, it is sufficient to show that every filter of $A$, is finitely generated. Let $G$ be an arbitrary filter of $A$, then we prove that $G$ is finitely generated. Let $H=\{ F_i \subseteq G: F_i$ be a finitely generated filter of $A\}$. For every ${ x}\in { G}$, we have, $\left\langle { x}\right\rangle \subset { G}$, so $H\ne \emptyset$. According to the previous argument, $H$ has a maximal element like ${ F}_{{ 1}}$. We prove ${ F}_{{ 1}} { =}G$  otherwise, if ${ F}_{{ 1}} \ne { G}$, since ${ F}_{{ 1}} \in { H}$, so ${ F}_{{ 1}} \subset G$ and there exists $x \in G-F_1$. Since ${ F}_{{ 1}} $ is finitely generated, then  ${ F}_{{ 1}} { =}\left\langle { x}_{{ 1}}, { x}_{{ 2}}, \ldots, { x}_{{ n}} \right\rangle $, for some $x_1,x_2,\dots,x_n\in A$. Thus ${ F}_{{ 1}} \subset { F}_{{ 2}} $=$\left\langle { x}_{{ 1}}, { x}_{{ 2}}, \ldots, { x}_{{ n}} {, }x\right\rangle \subset { G}$, so ${ F}_{{ 2}} \in { H}$ and $F_{{ 1}} \subset $ ${ F}_{{ 2}} $, which is a contradiction. Hence $G$ is a finitely generated filter of $A$ and by Theorem 2.3, $A$ is a Noetherian $BL$-algebra. \\
		
 \textbf{Theorem 3.4} Let $A$ be a $BL$-algebra. Then $A$ is a semilocal if and only if $A/M(A)$ is a Boolean Artinian $BL$-algebra.

\textbf{Proof.} Let $A$ be a semilocal $BL$-algebra. We show that by Definition 2.10, any properly descending chain of Boolean deductive system in $A/M(A)$ is finite. Suppose $M_1, M_2, M_3,\dots, M_n$ be the $n$ disjoint maximal deductive systems of $A$. Then by Definition 2.7, $M(A)=\cap \{M:$ $M$ is a maximal deductive system of $A\}$, i.e., $M(A)=\bigcap _{k=1}^n M_k$. By Proposition 2.18, $A/M(A)$ is isomorphic to $\mathcal{A}= \prod_{k=1}^n A/M_k$. Also by Proposition 2.8, every properly descending chain of deductive system of $\mathcal{A}$ is finite. Thus $A/M(A)$ is Boolean Artinian.

Conversely, Let $A/{M(A)}$ be  Boolean Artinian, then any properly descending chain of deductive system in $A/M(A)$ is finite. So any properly descending chain of deductive system of $A$ is finite, thus, $A$ is a semilocal $BL$-algebra.  If $A$ contains infinite many maximal deductive system like $M_1, M_2, M_3,\ldots, $ then $M_1\supseteq ( M_1 \cap M_2)\supseteq (M_1 \cap M_2\cap M_3)\supseteq \dots$ , is an infinite properly descending chain of deductive system generating an infinite properly descending chain of deductive system $M_1/M(A)\supseteq (M_1 \cap M_2)/M(A)\supseteq (M_1 \cap M_2\cap M_3)/M(A)\supseteq \dots$, of deductive system into $A/M(A)$, this is a contradiction. Since $A/M(A)$ is Boolean Artinian, so $A$ contains only finite many disjoint maximal deductive system, i.e., $A$ is a semilocal $BL$-algebra. \\

By Theorem 3.4, the following conditions are equivalent:\\

\textbf{Corollary 3.5} Let $A$ be a $BL$-algebra. Then the following conditions are equivalent:

$(i)$	$A$ is a semilocal and $A$ has the $n$ disjoint maximal deductive systems;

$(ii)$ $A/M(A)$ is Boolean Artinian;

$(iii)$ $A/M(A)$ is a semilocal $BL$-algebra.

\textbf{Proof. }
$(i)\Rightarrow (ii)$  Let $A$ be a semilocal and $A$ has the $n$ disjoint maximal deductive systems. Then, by Theorem 3.4, $A/M(A)$ is Boolean Artinian.

$(ii)$ $\Rightarrow$ $(i)$  Let $A/M(A)$ be Boolean Artinian. Then, by Theorem 3.4, $A$ is a semilocal $BL$-algebra and by Definition 2.9, $A$ has the $n$ disjoint maximal deductive systems.

$(i)$ $\Rightarrow$ $(iii)$  Let $A$ be a semilocal and $A$ has the $n$ disjoint maximal deductive systems. Suppose $M_1, M_2, M_3,\ldots, M_n$ be the $n$ disjoint maximal deductive systems of $A$. Then by Definition 2.7, $M(A)=\cap \{M:M$ is a maximal deductive system of $A\}$, i.e., $M(A)=\bigcap _{k=1}^n M_k$. By Proposition 2.18, $A/M(A)$ is isomorphic to $\mathcal{A}= \prod_{k=1}^n A/M_k$. Also by Proposition 2.8, every properly descending chain of deductive systems of $\mathcal{A}$ is finite. Thus $A/{M(A)}$ is a semilocal $BL$-algebra. \\
$(iii)$ $\Rightarrow$ $(ii)$  Let $A/M(A)$ be a semilocal $BL$-algebra. Suppose $M_1 /M(A)\supseteq M_2/M(A)\supseteq M_3/M(A)\supseteq  \dots$, be properly descending chain of deductive system of a semilocal $BL$-algebra $A/M(A)$ in which $M_1, M_2, M_3, \ldots$, are maximal deductive systems of $A$. Since $A/M(A)$ is a semilocal, so it contains only finite many disjoint maximal deductive systems. Hence for some $n\geq1$, $M_n/M(A)= M_{n+1}/M(A)$. Thus $A/M(A)$ is a Boolean Artinian $BL$-algebra. 	\\	
			
	\textbf{Theorem 3.6} Let $A$ be a $BL$-algebra and $A\ne \emptyset$. Then the set of prime ideals of $A$ has minimal element with respect to inclusion.
	
	\textbf{Proof.} Put $X=\{P:P$ is a prime ideal in $A\}$. Since $A\ne \emptyset, X\ne \emptyset$, consider $(X, \supseteq )$. Suppose ${P_i }$ be a chain in $(X, \supseteq )$. Put $P=\cap _i P_i$. Clearly, $P$ is an ideal of $A$. We prove $P$ is prime. Let $a\wedge b\in P$ and $a\notin P, ~b\notin P$. Then there exist $i, j$ such that $a\notin P_i, ~b\notin P_j$. Since ${P_i }$ is a chain, so $P_i \supseteq P_j$, also $a\notin P_j$. Hence $a\wedge b\notin P_j$, then $a\wedge b\notin P=\cap _i P_i$ which is a contradiction, so $P$ is prime. Therefore $X$ has a minimal element with respect to inclusion. \\
	
	
	\textbf{Theorem 3.7} Let $A$ be a $BL$-algebra. Then every finitely generated ideal in $A$ is principal.
	
	\textbf{Proof.} Let $F$ be a finitely generated ideal of $A$, then $F= (x_1, \ldots, x_n ]$ for some $x_1, \ldots, x_n\in A$. By induction on $n$, we complete the proof. If $n=2$, the claim is true by hypothesis. Now assume it is true for $n=k$ and set $(x_1, \ldots, x_n ] = (x]$. Let $n=k+1$ then $(x_1, \ldots, x_k, x_{k+1} ] \subseteq ( (x]\cup ( x_{k+1} ]]$, by Proposition 2.12, we have $( ( x]\cup (x_{k+1} ]] = ((\bar{x} \odot \bar{x}_{k+1} )^-] $.
		
		Conversely, since $( x], (x_{k+1} ] \subseteq ( x_1, …, x_k, x_{k+1} ]$, then $((\bar{x} \odot \bar{x}_{k+1} )^- ] =(x]\cup ( x_{k+1} ]]\subseteq ( x_1, \ldots, x_k, x_{k+1} ]$, so $( x_1, \ldots, x_k, x_{k+1} ] =((\bar{x} \odot \bar{x}^-_{k+1})^- ]$ is a principal ideal of $A$, and the claim holds for all $n\in \mathbb{N}$. \\
	
	
	\textbf{Theorem 3.8} If Every ideal of $BL$-algebra $A$ is principal, then $A$ is a co–Noetherian $BL$-algebra.
	
	\textbf{Proof.} Let $I_1\subseteq I_2\subseteq \dots\subseteq I_n\subseteq\dots $, be an increasing chain of ideals of $A$. Put $I=I_1\cup I_2\cup \dots, $ it is clear that $I$ is an ideal of $A$. By hypothesis $I = (x]$ for some $ x \in A$. So there exists $n\in\mathbb{ N}$ such that $x \in I_n$. Therefore, $I_i=I_n$, for all $i\geq n$. Hence $A$ is a co–Noetherian $BL$-algebra. \\
	
	\textbf{ Proposition 3.9} If $A$ is a Noetherian $BL$-algebra. Then $(Spec(A), T_A)$ is a Noetherian topological space, where $D(F)=\{P\in Spec(A): F\nsubseteq P\}$ and $T_A=\left\{D\left(F\right):F\in F_i\left(A\right)\right\}$.
	
	\textbf{Proof. } From [3], we deduce, $T_A=\left\{D\left(F\right):F\in F_i\left(A\right)\right\}$ is a topological space. Let $F_1\subseteq F_2\subseteq \dots $, be a chain of $A$. By Theorem 2.17 $(i)$ clearly, $D(F_1)\subseteq D(F_2)\subseteq \dots $, is a chain of $T_A$. Since $A$ is Noetherian, so there exists $i\in\mathbb{N}$, such that for all $n\ge i\, \ F_i=F_{n\ }.$ Then by Theorem 2.17 $(ix)$, $D(F_i)=D\left(F_n\right)$, for all $n\ge i$ and hence $(Spec(A), \ T_A)$ is Noetherian topological space. \\



\noindent \textbf{Proposition 3.10} Let $A$ be a Boolean Artinian $BL$-algebra, $B$ be a $BL$-algebra and $f:A\longrightarrow B$ be an onto $BL$-homomorphism. Then $f(A)=B$ is a Boolean Artinian $BL$-algebra.

\noindent \textbf{Proof.} It is easy to see that for any deductive system $D$ of $B$, $f^{-1}(D)$ is a deductive system of $A$. Let $D_1\supsetneq D_2\supsetneq \dots \supsetneq {D}_n\supsetneq \dots, $ be a properly descending chain of Boolean deductive systems of $B$, then $f^{-1}(D_i)${, }$ i\in {\rm }{\mathbb N}$ are Boolean deductive systems of $A$. Since $f$ is onto, $ f^{-1}(D_1)\supsetneq f^{-1}(D_2)\supsetneq \dots \supsetneq f^{-1}(D_n)\supsetneq \dots $, is a properly descending chain of Boolean deductive system of $A$. By hypothesis, $A$ is Boolean Artinian, then there exists a $n\in {\rm }{\mathbb N}$ such that, $f^{-1} (D_i)=f^{-1}(D_n)$, for all $i\ge n$. Since $f$ is an onto $BL$-homomorphism, so we get $f( f^{-1} ( D_{i}))= {f(f^{-1} (D}_n))$  and $D_i=D_n$, for all $i\ge n$. Thus $B$ is a Boolean Artinian $BL$-algebra.\\

\noindent \textbf{Proposition 3.11} Let $A$ be a Boolean Artinian $BL$-algebra and $f:A\longrightarrow A $ be a one to one $BL$-homomorphism. Then $f$ is an onto $BL$-homomorphism.

\noindent \textbf{Proof.} Let $f$ is not an onto $BL$-homomorphism of $BL$-algebras, i.e., $A\supsetneq f( A )${. }Since $f$ is one to one, so $f(A)\supsetneq f^2(A)$ and hence $f^{n-1}(A)\supsetneq f^n(A)$ for all $n\ge 2$. This means that $A\supsetneq f ( A )\supsetneq f^2 ( A )\supsetneq \dots \supsetneq f^{n-1}\left(A\right)\supsetneq f^n\left(A\right)\supsetneq \dots, $ is a properly descending chain of Boolean deductive systems of $A$. This chain is not stationary, because if there exists $k\in {\rm }{\mathbb N}$ such that $f^{k+1}(A) =f^k(A)$, then by the injectivity of $f$, there exists a map $g: A\rightarrow A, $ $g(f(A))=I_A$, thus $g ( f^{k+1}(A ))=g(f^k\left(A\right))$, i.e., $f^{k}\left(A{\rm }\right){\rm =}{\rm }f^{k-1}\left(A\right)$, by continuing this procedure, we get, $A = f ( A )$. This is a contradiction, hence $A = f(A)$ and $f$ is an onto $BL$-homomorphism. \\

\noindent \textbf{Proposition 3.12 }Let $A$ be a Boolean Artinian $BL$-algebra and $D$ be a deductive system of $A$. Then $\dfrac{A} {D}$ is a Boolean Artinian $BL$-algebra.

\noindent \textbf{Proof.}~Let $\dfrac{D_1} {D}\supsetneq \dfrac{D_2} {D}\supsetneq \dots \supsetneq \dfrac{D_n} {D}\supsetneq \dots, $ be a properly descending chain of Boolean deductive system of $\dfrac{A} {D}$. Then $D_1\supsetneq D_2\supsetneq \dots \supsetneq {D}_n\supsetneq \dots, $ is a properly descending chain of Boolean deductive system of $A$. Since $A$ is a Boolean Artinian $BL$-algebra, there exists a $n\in {\rm }{\mathbb N}$ such that for all $i\ge n$, $D_i=D_n,$ so for all $i\ge n$, $\dfrac{D_i} {D} =\dfrac{D_n} {D}$. Thus $\displaystyle\frac{A} {D}$ is Boolean Artinian. \\

\noindent \textbf{Proposition 3.13} Let $A$ be a $BL$-algebra. Then $A$ is a Boolean Artinian $BL$-algebra if and only if every non empty set of deductive systems of $A$ has a minimal element.

\noindent \textbf{Proof. }Let $A$ be a Boolean Artinian $BL$-algebra and $G$ be a nonempty set of deductive systems of $A$ that does not have a minimal element. There exists a $D_1\in $ $G$, since $G$ is a non empty set. Now as $G$ does not have a minimal element, there exists $D_2 \in G$ such that $D_1\varsupsetneq D_2$. Continuing this method, we have $D_1\supsetneq D_2\supsetneq D_3\supsetneq \dots $, which is a properly descending chain of Boolean deductive systems of $A$. This chain is not stationary, so which is a contradiction. Thus $G$ has a minimal element.

\noindent Conversely, suppose $D_1\supsetneq D_2\supsetneq D_3\supsetneq \dots $, is a properly descending chain of Boolean deductive systems of $A$. Put $G=\{D_i: i\in {\rm }{\mathbb N} \}$. Since $G$ is a non empty set, $G$ has a minimal element, like $ {D}_n$. Hence for all $i\ge n$, $D_i=D_n$ and $A$ is a Boolean Artinian $BL$-algebra. \\

\noindent \textbf {Proposition 3.14} Let $A$ be a Boolean Artinian $BL$-algebra. Then the set of all maximal deductive systems of $A$ is finite.

\noindent \textbf{Proof.} Put $G =\{D\in \texttt{D}(A): D$ is the intersection of finitely many maximal deductive systems of $A\}$. If $Max(A)$ is a non empty set, then $G$ is also a non empty set. Thus by Proposition 3.13, $G$ has a minimal element $D_1$. So there exist $M_1, M_2,\dots, M_n$ of the set of all maximal deductive systems of $A$ such that $D_1=M_1\cap M_2\cap \dots \cap M_n$. Suppose $M$ is an element of the set of all maximal deductive systems of $A$. Then $M\cap D_1\subseteq D_1$, $M\cap D_1= M\cap M_1\cap M_2\cap \dots \cap M_n\in G$ and $D_1$ is a minimal element of $G$, so $M\cap D_1= D_1${, Hence }$D_1=$ $M_1\cap M_2\cap \dots \cap M_n\subseteq M$. Since any deductive system is a filter and every maximal filters is prime, so $M$ is in $Spec(A)$, thus there exists $ i\in \mathbb{N}$, such that $M_i\subseteq M$. Now as $M$, $M_i$ are elements of the set of all maximal deductive systems of $A$, we get $M_i=M. $ Hence the set of all maximal deductive systems of $A$ is finite. \\

\noindent \textbf{Proposition 3.15} Let $A$ and $B$ be two local $BL$-algebras and $f:A\longrightarrow B$ be a $BL$-homomorphism. Then $f(A)$ is a Boolean Artinian $BL$-algebra.

\noindent \textbf{Proof.} First we show that $A$ is a Boolean Artinian $BL$-algebra. Suppose $D_1\varsupsetneq D_2\varsupsetneq \dots \varsupsetneq {D}_n\varsupsetneq \dots,$ be a properly descending chain of Boolean deductive system of a local $BL$-algebra $A$. We claim that, there exists a $n\in {\rm }{\mathbb N}$ such that for all $i\ge n$, $D_i=D_n$. It is clear that, $D_1\varsupsetneq D_2\varsupsetneq \dots \varsupsetneq {D}_n\varsupsetneq \dots, $ generates a properly descending chain ${\left\langle D_1\cup M\left(A\right)\right\rangle {\rm }}/{M\left(A\right)}\varsupsetneq {\left\langle D_2\cup M\left(A\right)\right\rangle {\rm }}/{M\left(A\right)}\varsupsetneq  \dots$, of deductive system in ${A}/{M\left(A\right)}$, where $M(A)=\bigcap \{M:M$ is a maximal deductive system of $A\}$. Since $A$ is a local $BL$-algebra, so ${A}/{M\left(A\right)}$ is semilocal and for some $ i\ge n$, we have ${\left\langle D_i\cup M\left(A\right)\right\rangle {\rm }}/{M\left(A\right)}={\left\langle D_n\cup M\left(A\right)\right\rangle {\rm }}/{M\left(A\right)}$. We show that $D_i=D_n$. Suppose $a\in {D}_i,$ then $a\in D_i\cup M\left(A\right)\subseteq \left\langle D_i\cup M\left(A\right)\right\rangle $ and $a\in \left\langle D_i\cup M\left(A\right)\right\rangle =\left\langle D_n\cup M\left(A\right)\right\rangle $, $a\in \left\langle D_n\cup M\left(A\right)\right\rangle $, so for some $b\in D_n$ and $c\in M(A)$, we have $b\odot c\le {a}$. Thus $b\le c\longrightarrow a$ and since $b\in D_n$, $D_n$ is a deductive system, hence $c\longrightarrow a\in D_{n}$. Since $M(A)$ is a properly deductive system, $c\in M(A)${, } and by Definition 2.7, $M(A)=\left\{x\in A: {x}^n>0\ for\ all\ n\ge 1\right\}$, so $\overline{c}\notin M(A)$, $ ( {\overline{c} )}^k= 0$, for some $k\ge 1$. Since $D_n$ is Boolean, then $c\vee \overline{c}\in D_n$, thus $( {{\rm c}\vee {\rm }\overline{c})}^k\in D_n $ and $c\vee 0=c$, $({{\rm c}\vee 0)}^k= {c}^k$, so $c^k=( {{\rm c}\vee \overline{c})}^k$, i.e., $c^k\in D_n.$  We know that $c^k\le c $ implies $c\longrightarrow a\le {c}^k\longrightarrow a$, therefore, from $c\longrightarrow a\in D_n $, We conclude that $a\in D_n$, i.e., $D_i=D_n$, for all $i\ge n$ and $A$ is Boolean Artinian. Since $B$ is a local $BL$-algebra, so $B$ is Boolean Artinian, and by Proposition 3.10, since $A$ and $B$ are Boolean Artinian, so $f(A)$ is a Boolean Artinian $BL$-algebra. \\

\noindent \textbf{Corollary 3.16 }If $A$ is a local $BL$-algebra and $f:A\rightarrow A $ be a one to one $BL$-homomorphism. Then $f$ is an onto $BL$-homomorphism.

\noindent \textbf{Proof.} Since $A$ is local $BL$-algebra, so by Proposition 3.15, $A$ is also Boolean Artinian. Thus by Proposition 3.11, $f$ is an onto $BL$-homomorphism. \\

\noindent \textbf{Corollary 3.17 } Let $A$ and $B$ be two $BL$-algebra and $f:A\longrightarrow B $ be an onto $BL$-homomorphism. If $A$ is a local $BL$-algebra. Then $B$ is a Boolean Artinian $BL$-algebra.

\noindent \textbf{Proof.} Since $A$ is a local $BL$-algebra, so by Proposition 3.15, $A$ is also Boolean Artinian, and by Proposition 3.10, $B$ is a Boolean Artinian $BL$-algebra. \\

\noindent \textbf{Corollary 3.18} Let $A$ be a local $BL$-algebra and $D$ be a deductive system of $A$. Then $ \dfrac{A} {D}$ is a Boolean Artinian $BL$-algebra.

\noindent \textbf{Proof.} Since $A$ is a local $BL$-algebra, so by Proposition 3.15, $A$ is Boolean Artinian and by Proposition 3.12, $\displaystyle \frac{A} {D}$ is a Boolean Artinian $BL$-algebra. \\

\noindent \textbf{Corollary 3.19} Let $A$ be a local $BL$-algebra. Then every non empty set of deductive systems of $A$ has a minimal element.

\noindent \textbf{Proof.} Since $A$ is local $BL$-algebra, so by Proposition 3.15, $A$ is Boolean Artinian, and by Proposition 3.13, $A$ is a Boolean Artinian $BL$-algebra if and only if every non empty set of deductive systems of $A$ has a minimal element. \\

\noindent \textbf{Corollary 3.20} Let $A$ be a local $BL$-algebra. Then the set of all maximal deductive systems of $A$ is finite.

\noindent \textbf{Proof.} Since $A$ is local $BL$-algebra, so by Proposition 3.15, $A$ is Boolean Artinian and by Proposition 3.14, the set of all maximal deductive systems of $A$ is finite. \\

\noindent \textbf{Proposition 3.21} Let $A$ and $B$ be two local $BL$-algebras and $h:A\longrightarrow B$ be an one to one $BL$-homomorphism. If $\left\langle h (D_1 )\right\rangle =\left\langle h (D_2 )\right\rangle $, then $D_1=D_2$, for all deductive systems $D_1, D_2$ of $A$.

\noindent \textbf{Proof.} Let $D_1$ and $D_2$ be two deductive systems of $A$ and $\left\langle h (D_1 )\right\rangle =\left\langle h (D_2 )\right\rangle $. Then for any $x\in D_1$, $h(x)\in \left\langle h (D_1 )\right\rangle =\left\langle h (D_2 )\right\rangle$, by Theorem 2.15, there exists $n\in {\rm }{\mathbb N}$ such that $(h(x_1)\odot h(x_2)\odot \dots \odot h(x_n))\longrightarrow h(x) = 1$, since $h$ is a $BL$-homomorphism, then $h(x_1\odot x_2\odot \dots \odot x_n) = (h(x_1)\odot h(x_2)\odot \dots \odot h(x_n))\longrightarrow h(x) = 1$, i.e., $h((x_1\odot x_2\odot \dots \odot x_n)\longrightarrow x) = 1$, $h(1) = 1$ and by assumption, $(x_1\odot x_2\odot \dots \odot x_n)\longrightarrow x = 1$. Since $x_1, $ $x_2, \dots, x_n\in D_1$ and by Theorem 2.15, $x\in \left\langle D_2\right\rangle = D_2$, so $D_1\subseteq D_2$.

\noindent Suppose $x\in D_2$, then $h (x)\in \left\langle h (D_1 )\right\rangle =\left\langle h (D_2 )\right\rangle $, by Theorem 2.15, there exists $n\in {\rm }{\mathbb N}$ such that $(h(x_1)\odot h(x_2)\odot \dots \odot h(x_n))\longrightarrow h(x) = 1$, since $h$ is a $BL$-homomorphism, then $h(x_1\odot x_2\odot \dots \odot x_n) = (h(x_1)\odot h(x_2)\odot \dots \odot h(x_n))\longrightarrow h(x) = 1$, i.e., $h((x_1\odot x_2\odot \dots \odot x_n)\longrightarrow x) = 1$, $h(1) = 1$ and by assumption, $(x_1\odot x_2\odot \dots \odot x_n)\longrightarrow x = 1$. Since $x_1, x_2, \dots, x_n\in D_2$ and by Theorem 2.15, $x\in \left\langle D_1\right\rangle = D_1$, so $D_2\subseteq D_1$. Hence $D_1=D_2$, for all $D_1, D_2\in D (A). $\\

\noindent\textbf{Corollary 3.22} Let $A$ be a $BL$-algebra such that every ideals of $A$ is finitely generated. If $B$ is sub $BL$-algebra of $A$, and $B$ is a co-Noetherian $BL$-algebra, then every sub $BL$-algebra $S$ between $B$ and $A$ ($B\subseteq S\subseteq A$) is also a co-Noetherian $BL$-algebra.

	\noindent\textbf{Proof.} Let $S$ be every sub $BL$-algebra between $B$ and $A$, i.e., $B\subseteq S\subseteq A$. By Definition 2.5, since every ideal of $A$ is finitely generated, so $A$ is a co-Noetherian $BL$-algebra and every ascending chain in $S$, stops in $A$, thus $S$ is co-Noetherian.

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%354.
		
		



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\end{center}



{\small

\noindent{\bf Jamal Kazemiasl}

\noindent  Ph.D student,

\noindent Department of Mathematics,

\noindent Shahrekord Branch,

\noindent Islamic Azad University,

\noindent Shahrekord,

\noindent Iran.

\noindent E-mail: Kazemiasl.j@gmail.com}\\

{\small
\noindent{\bf  Farhad Khaksar Haghani (Corresponding Author) }

\noindent Associate Professor of Mathematics,

\noindent Department of Mathematics,

\noindent Shahrekord Branch,

\noindent Islamic Azad University,

\noindent Shahrekord,

\noindent Iran.

\noindent E-mail: Haghani1351@yahoo.com}\\

{\small
\noindent{\bf  Shahram Heidarian }

\noindent Assistant Professor of Mathematics,

\noindent Department of Mathematics,

\noindent Shahrekord Branch,

\noindent Islamic Azad University,

\noindent Shahrekord,

\noindent Iran.

\noindent E-mail:Heidadrianshm@gmail.com}\\


\end{document} 