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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