section{$f{L}$-$f{VL}$-algebras, Let A be an $ell$-$f{VL}$-algebra.A non-empty

section{$f{L}$-$f{VL}$-algebras, $L$-Boolean Systems, $L$-Boolean Space and their Categorical interconnections}
subsection{f{L-VL-algebras}}
$L$-valued logic $L$-$f{VL}$ is a many-valued logic and $ell$-$f{VL}$-algebras defined in cite{m} as an algebraic semantics for $ell$-$f{VL}$, which are both sound and complete.
egin{defn}(cite{7})
$ell$-$f{VL}$-algebra homomorphism is a function between two $ell$-$f{VL}$-algebras which preserves the operations $(vee, wedge,
ightarrow, T_a(a in ell), 0,1)$.

end{defn}
egin{defn}(cite{7})
Let A be an $ell$-$f{VL}$-algebra.A non-empty subset $F$ of $A$ is called an $ell$-filter iff $F$ is a filter of lattices which is closed under $T_1$.Let $P$ be a proper $ell$-filter of $A$.
egin{enumerate}
item P is a prime $ell$-filter of $A$ iff for any $rin ell$, $T_r(xvee y)in P$, then there exist $r_1,r_2in ell$ with $r_1vee r_2=r$ such that $T_{r_1}(x) in P$ and $T_{r_2}(y)in P$.
item P is an ultra $ell$-filter of $A$ iff $forall rin A$ $exists rin ell$ ,$T_r(x)in P$.
item $P$ is a maximal $ell$-filter iff $P$ is maximal with respect to inclusion.
end{enumerate}
end{defn}
egin{prop}(cite{7}
egin{enumerate}
item Let $A$ be an $ell$-$f{VL}$-algebra.For any two distinct members $x,y$ of $A$ ,there exist $rin ell$ and a prime $ell$-filter $P$ of $A$ such that $T_r(x)in P$ and $T_r(y)
otin P$.
item For a prime $ell$-filter $P$ of an $ell$-$f{VL}$-algebra $A$,define $v_P:A
ightarrow ell$ by $v_P(x)=q Leftrightarrow T_q(x)in P$.Then,$v_P$ is a homomorphism of $ell$-$f{VL}$-algebras.
item Let $A$ be an $ell$-$f{VL}$-algebra.A bijective mapping exists from the set of all prime $ell$-filters of $A$ to the set of all homomorphisms from $A$ to $ell$.

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end{enumerate}
end{prop}

The spectrum of an $ell$-$f{VL}$-algebra $A$ is designated by $Spec_ell(A)$, and is defined as follows.
egin{defn}(cite{7})
Let $A$ be an $ell$-$f{VL}$ algebra.For a subalgebra $ell_1$ of $ell$, $Spec_{ell_1}(A)=$${ell$-$f{VL}$-algebras homomorphism $f:A
ightarrow ell_1}$.
end{defn}
section{Categories and their Functorial relationships}
$ell$-f{Boolean systems}
egin{defn}
An $ell$-Boolean system is a triple $(X,A,models)$, where $X$ is a non-empty set, $A$ is an $ell$-$f{VL}$- algebra and $models$ is an $ell$-valued satisfaction relation on $(X,A)$, i.e., $models :X imes A
ightarrow ell$ is a mapping such that
egin{enumerate}
item if $A_1$ is a subset of $A$ ,then $displaystyle models(x,igvee_{a in A_1})=igvee_{ain A_1}models(x,a)$,
hspace{0.5in} if $A_1$ is a finite subset of $A$, then $displaystylemodels(x,igwedge_{ain A_1})=igwedge_{ain A_1}models(x,a)$
item if $x_1
eq x_2$ in $x$ then $displaystylemodels(x,a_1)
eq models(x_2,a)$, for some $ain A$.
item $models(x,a
ightarrow b)=models(x,a)
ightarrow models(x,b)$.
item $models(x,T_r(a))=T_r(a)$,$ain A$ and $r in ell$.
item $models(x,0)=0$,$models(x,1)=1$.Here 1,0 are respectively the top-element and bottom-element.

end{enumerate}

end{defn}

egin{defn}We define a category $ell$-{f BSYM} as follows.
(1) Objects are all $ell$-Boolean systems $(X,A,models)$.
(2) Arrows are all continuous maps $(psi_1,psi_2):(X,A,models_1)
ightarrow (y,B,models_2)$, where
egin{center}
egin{itemize}
item$psi_1:X
ightarrow Y$ is a set map.
item $psi_2:B
ightarrow A$ is a homomorphism of $ell$-{f{VL}}-algebras.
item $models_1(x,psi_2(b))=models_2(psi_1(x),b)$
end{itemize}
end{center}
(3) For each object $P=(X,A,models)$ ,the identity arrow $I_P:P
ightarrow P$ is the pair $(I’,I”)$ such that
egin{center}
$I’:X
ightarrow X$
$I”:A
ightarrow A$.
end{center}
(4) For a given $ell$-Boolean systems $P=(X,A,models_1)$, $Q=(Y,B,models_2)$, $R=(Z,C,models_3)$ let $(psi_1,psi_2):P
ightarrow Q$ and $(phi_1,phi_2):Q
ightarrow R$ be continuous maps.Composition defined as $ (phi_1,phi_2)circ(psi_1,psi_2):P
ightarrow R$ such that
egin{center}
$phi_1circpsi_1:X
ightarrow Z$
$psi_2circphi_2:C
ightarrow A$.
end{center}

end{defn}