Copyright © 2018 Authors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. International Journal of Engineering & Technology, 7 (4.30) (2018) 160-164 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET Research paper Fuzzy Finite Switchboard Automata with Complete Residuated Lattices Nur Ain Ebas 1 , Nor Shamsidah Amir Hamzah 1 *, Kavikumar Jacob 1 , Mohd Saifullah Rusiman 1 1 Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia, Kampus Pagoh, 84600, Pagoh, Muar, Johor, Malaysia *Corresponding author E-mail:shamsida@uthm.edu.my Abstract The theory of fuzzy finite switchboard automata (FFSA) is introduced by the use of general algebraic structures such as complete residuated lattices in order to enhance the process ability of FFSA. We established the notion of homomorphism, strong homomorphism and reverse homomorphism and shows some of its properties. The subsystem of FFSA is studied and the set of switchboard subsystem- forms a complete ℒ -sublattices is shown. The algorithm of FFSA with complete residuated lattices is given and an example is provided. Keywords: Complete Residuated Lattices; Fuzzy Finite Automata; Fuzzy Finite Switchboard Automata; Switchboard Subsystems. 1. Introduction A finite state machine is a mathematical computation model that is used to design the computer programs along with sequential logic circuits. Many researchers have studied the idea of finite automata since 1940. The theory of automata had investigated by Rabin and Scott [7]. Wee [23] is the first researcher who introduced the concept of the theory of automata in the fuzzy environment. In general fuzzy finite state machine (FFSM) or fuzzy finite automata (FFA) has membership grades in an interval [0,1] but there is a possibility to extended the membership values into more general algebraic structures. Qiu studied those so-called theories and their characterizations where he considered its membership grades under the fact of complete residuated lattices [17,20]. In the following year, Qiu extended his research into specific type of automata which are pushdown automata, turning machine and reduction and minimization [5,9,17]. Many researchers studied on fuzzy finite automata with membership value in a Complete Residuated Lattices (CRL) [2,4,9,11,13,14,15,17,19,20,21,22]. As a continuation of the FFA, the concept of fuzzy finite switchboard state machine (FFSSM) that is made up of switching and commutative state machines has been studied by Sato and Kuroki [3,6]. This classified fuzzy finite automaton has a mechanism that will act as a controller during the transition between the current state and next. In this research, a complete residuated lattice is chosen because it offers the general algebraic structures associated with several important logics [1,12, 17, 18]. According to literature, the CRL has not been applied to FFSA. Therefore, in this research, the theory of FFSA is extended to a more comprehensive structure by considering the membership values in a complete residuated lattice. 2. Preliminaries “An algebraic structure with strong connections to mathematical logic is known as a residuated lattice. Definition 2.1 [24] The algebra  = (,∧,∨,⊗, →, , ) should satisfying three conditions: a) (,∧,∨ ,0,1) is a lattice with the least element is 0 and the greatest element is 1 b) (,⊗,∨) is a commutative monoid with the unit 1, c) ⊗ and → form an adjoint pair. For example, they satisfy the adjunction property: for all , ,  ∈ ,  ⊗  ⇔  ≤  → . Let ℒ = (,∧,∨,⊗, → ,0,1) where ℒ is called complete residuated lattice if (,∧,∨ ,0,1) is a complete lattice, ⊗ is called a multiplication, → is a residuum, ∧ and ∨ is supremum and infimum respectively. Multiplication, ⊗ and residuum, → are intended for modeling the conjunction and implication of the corresponding logical calculus. Meanwhile, Supremum ∨ and infuimum ∧ are intended to model the general and existential quantifier. The notion ⇔ (biimplication) can be written as ( → ) ∧ ( → ),  →  = min(1 −  + , 1) is a complete residuated lattice while  ⊗  = max( +  − 1, 0) is Standard Lukasiewcz algebra. Heyting algebra is a residuated lattice that satisfies ⊗ =∧. Meanwhile, the notion of Standard Godel algebra is  ⊗  = min(, ) and →=1 if ≤ and  otherwise is a Heyting algebra [13]. There are some properties of complete residuated lattice in the following lemma: Lemma 2.2 [17, 13] Let  be a complete residuated lattice. Then , ,  ∈  and {  } ∈ , {  } ∈ ⊆ the following properties hold: 1) ≤ if and only if →=1