## 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}