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The paper explores categorical interconnections between lattice-valued Relational systems and algebras of Fitting’s lattice-valued modal logic.We define lattice-valued boolean systems, and then we study co-adjointness, adjointness of functors.As a result, we get a duality for algebras of lattice-valued logic.Following this duality results, we establish a duality for algebras of lattice-valued modal logic.\

{f {Keywords:}}
Lattice-valued Boolean space, Lattice-valued topological systems, Algebras of Fitting’s lattice-valued modal logic, Adjoint, Coadjoint, natural duality.
section*{Introduction}
Vicker’s in cite{1} introduced the concept of topological systems in the work of topology via logic, which was further considered in cite{2}.Topological systems is a mathematical structure like as \$(X,A,models)\$,where \$X\$ is a non-empty set, \$A\$ is a frame,a complete distributive lattice and \$models\$ is a satisfaction relation on \$X imes A\$.This relation \$models\$ satisfies both join and finite meet interchange laws.\
The authors in cite{melton} demonstrated the idea of lattice-valued topological systems and considered the category \$f {L-TopSys}\$ from lattice-valued topological systems .They also explored categorical relation with the systems and spaces. Functorial relationship between the category of variable-basis topology and topological systems was shown in cite{D, R}, which was further studied in cite{27,28}.\
In cite{6} Fitting revealed the idea of \$L-valued\$ logic and \$L-valued\$ modal logic for a finite distributive lattice \$L\$, where \$L\$ is endowed with the truth constants.Several have been studied in different aspects of Fitting logics (cite{7,8,9,10}).Maruyama cite{11}defined the class of \$L-VL\$-algebras and the class of \$L-ML\$-algebras as an algebraic structure of Fitting’s \$L\$-valued logic and \$L\$-valued modal logic respectively.Consequently, a duality developed for \$L-VL\$ algebras and the category \$L\$-BS in cite{12}, which can be viewed as strong dality according to the theory of natural dualities cite{13}.Following the duality for \$L-f{VL}\$-algebras,he also generalized J\$acute{o}\$sson-Tarki style duality(e.g.,cite{G-haunssol}) for \$L-f{ML}\$-algebras.\
While studying cite{melton},we raised a question whether there is a systems which are categorically connected with L-VL-algebras and L-\$f{ML}\$-algebras.Our objective here is to define such systems and how these systems are categorically equivalent with the spaces(c.f.cite{may}).As a result ,we shall establish a duality for \$ell\$ -\$f{VL}\$-algebras and L-\$f{ML}\$-algebras.For the purpose of our proof, we may recall
The paper is organized as follows.In Section
ef{ 1} we recall some basic notions associated with this work.In Section 2

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