#### Abstract

We study the cubic finite switchboard state machines (cubic fssms) intensively. The notions of (resp., good, strong) subsystem of cubic fssm are introduced. We prove that the Cartesian composition of two (resp., good, strong) subsystems of cubic fssms is (resp., good, strong) subsystem of cubic fssm. Similarly, the product of two (resp., good, strong) subsystems of cubic fssms is (resp. good, strong) subsystem of cubic fssm. We define the homomorphism between two of cubic fssms and proved some related results. Many examples on each case are provided.

#### 1. Introduction

The idea of fuzzy set was first introduced by Zadeh [1]. He defined fuzzy sets which have been applied in daily real life as traffic monitoring systems, artificial intelligence, medicine, air conditions, washing machine, and computer science.

This paper is an extesion of cubic finite state machines which have been done by Abughazalah and Yaqoob [2]. Abdullah, Naz and Pedrycz [3] defined cubic finite state machines. Cartesian composition of automata was introduced first in [4]. Malik, Mordeson, and Sen [5–7] provided the concepts of both the product and the “Cartesian composition” of ffsm and they studied subsystem and strong subsystem of fuzzy finite state machines. Ebas, Hamzh, etc. [8] studied the algebraic properties of finite switchboard state machines and studied their properties. Constructions of product of finite switchboard state machines were given by Kumbhojkar and Chaudhari [9]. Mahmood and Khan [10] studied the notion of interval neutrosphic finite switchboard state machines. Hussain and Shabbir [11] introduced the concept of soft finite state machines and applied soft set theory to finite state machines.

The authors in [12–14] introduced intuitionistic fuzzy finite state machines and intuitionistic fuzzy finite switchboard state machines. Finite state machines in terms of bipolar fuzzy set were studied in [15] and some types of subsystem and strong subsystem in bipolar fuzzy set were introduced in [16]. Properties of bipolar fuzzy finite switchboard state machines and the concepts of bipolar submachine, bipolar connected, and bipolar retrievable are investigated in [17]. Jun et al. [18] studied the behaviour of cubic ideal (resp., ideal, ideal) of BCI-algebras. The concepts of (closed) cubic soft ideals in BCK/BCI- algebras are introduced and the relations between them are discussed in [19]. Liu et al. [20] studied the product of “Mealy- type fuzzy finite state machines”.

#### 2. Preliminaries

*Definition 1 (see [1]). *“A map is called a fuzzy set of .”

*Definition 2 (see [1]). *“An interval valued fuzzy set (briefly, IVF-set) on is defined as where , for all . Then the ordinary fuzzy sets and are called a lower fuzzy set and an upper fuzzy set of , respectively. Let Then where ”

*Definition 3 (see [15]). *“Let be a non-empty set. By a cubic set in we mean a structure in which is an IVF set in and is a fuzzy set in .”

*Definition 4 (see [3]). *“A cubic finite state machine (cubic fsm, shortly) is a triple , where and are finite non-empty sets, called the set of states and the set of input symbols, respectively, and is a cubic set in ”

“Let denote the set of all words of elements of of finite length. Let denote the empty word in and denote the length of for every See [3].”

*Definition 5 (see [3]). *“Let be a cubic fsm. Define a cubic set in by andfor all , and ”

Lemma 6 (see [3]). *“Let be a cubic fsm. Then for all and "*

#### 3. Cubic Finite Switchboard State Machines

*Definition 7. *A cubic fsm is said to be switching if it satisfiesfor all and

*Example 8. *Let and Let be a cubic subset in Then the cubic fsm is switching, as shown in Figure 1.

*Definition 9. *A cubic fsm is said to be commutative if it satisfiesfor all and

*Example 10. *Let and Let be a cubic subset in Then the cubic fsm is commutative, as shown in Figure 2.

The cubic fsm shown in Figure 1 is switching but not commutative becauseAlso, the cubic fsm shown in Figure 2 is commutative but not switching because

Proposition 11. *If is a commutative cubic fsm, then for all , and *

*Proof. *Let , , and We prove the result by induction on If , then ; hence andTherefore the result is true for Suppose that the result is true for That is, for all with , Let be such that Then andHence the result is true for This completes the proof.

*Definition 12. *A cubic fsm is called a cubic finite switchboard state machine (cubic fssm, shortly) if it is switching and commutative.

Proposition 13. *If is a cubic fssm, then for all and *

*Proof. *Let and . We prove the result by induction on If , then , hence andTherefore the result is true for Assume that the result is true for That is, for all with , , we have Let and be such that Then andThis shows that the result is true for This completes the proof.

Proposition 14. *If is a cubic fssm, then for all and *

*Proof. *Proof easily follows from Propositions 11 and 13.

*Definition 15. *Consider a cubic fssm and a cubic subset of Then is said to be a subsystem of if and only if for all and

If is a subsystem of , then we write simply for

*Example 16. *Let and Let and be a two cubic subsets in and , respectively. Then subsystem of the cubic fsm is shown in Figure 3.

Theorem 17. *Consider a cubic fssm and a cubic subset of Then is subsystem of if and only if for all and *

*Proof. *Suppose is a subsystem of . Let and We prove the result by induction on If , then Now, if then Now if , then andThus the result is true for Suppose the result is true for all such that , Let , , and Then andThe converse is obvious.

*Definition 18. *Consider a cubic fssm and a cubic subset of Then is said to be a good subsystem of if and only if for all and

*Example 19. *The good subsystem of a cubic fssm is shown in Figure 4.

Theorem 20. *Consider a cubic fssm and a cubic subset of Then is good subsystem of if and only if for all and *

*Proof. *The proof is similar to the proof of Theorem 17.

*Definition 21. *Consider a cubic fssm and a cubic subset of Then is said to be a strong subsystem of if and only if thenfor all and

*Example 22. *The strong subsystem of a cubic fssm is shown in Figure 5.

*Remark 23. *A strong subsystem may be or may be not a good subsystem.

Theorem 24. *Consider a cubic fssm and a cubic subset of Then is strong subsystem of if and only if thenfor all and *

*Proof. *Suppose that is a strong subsystem. We prove the result by induction on If , then Now, if then and Thus and Now if , then and Thus the result is true for Suppose the result is true for all such that , Let , , and Suppose that Then Thus there exists such that and Hence and Thus Now suppose that Then Thus there exists such that and Hence and Thus The converse is obvious.

Theorem 25. *Let be a cubic fssm. Let and be two subsystems (resp., good subsystems, strong subsystems) of Then the following conditions hold.**(i) is a subsystem (resp., good subsystem, strong subsystem) of **(ii) is a subsystem (resp., good subsystem, strong subsystem) of *

*Proof. *Suppose that and be two subsystems of Then alsofor all and

(i) Now to prove that is a subsystem of It is enough to prove that andNowand Hence this shows that is a subsystem of .

(ii) Now to prove that is a subsystem of It is enough to prove that andNowandHence this shows that is a subsystem of . The other cases can be seen in a similar way.

#### 4. Products of Subsystems of Cubic fssm

In [2], the authors constructed different types of products of subsystems of cubic fsms. Here in this section we will provide some results related to products of subsystems of cubic fssm.

*Definition 26. *Let and be two cubic fssms and let The Cartesian composition of and is denoted byand is defined as follows:

(i)for all , and .

(ii)for all , and

Proposition 27. *The “Cartesian composition” of two cubic fssms is a cubic fssm.*

*Proof. *It is straightorward.

*Definition 28. *Let and be the subsystems of cubic fssms and , respectively, and let The “Cartesian composition” of and is denoted by and is defined as follows:

(i)for all .

(ii)for all , and .

(iii)for all , and

Proposition 29. *Let and be two subsystems (resp., good subsystems, strong subsystems) of cubic fssms and , respectively. Then is a subsystem (resp., good subsystem, strong subsystem) of cubic fssm *

*Proof. *t is straightforward.

Theorem 30. *If is a good subsystem of cubic fssm , then at least or must be good subsystem.*

*Proof. *Suppose that and are not good subsystems. Then there exist , , , and such that alsoNow by the definition andNowandThis shows that is not good subsystem, a contradiction. This completes the proof.

The following examples show that if is a good subsystem of and is not a good subsystem of , then may or may not be a good subsystem.

*Example 31. *Consider is a good subsystem and is not a good subsystem as shown in the Figures 6 and 7.

The Cartesian composition of and is shown in Figure 8. It is clear from Figure 8 that is a good subsystem.

*Example 32. *Consider is a good subsystem and is not a good subsystem as shown in Figures 9 and 10.

The Cartesian composition of and is shown in Figure 11. It is clear from Figure 11 that is not a good subsystem

Now we can state a new proposition without its proof.

Proposition 33. *Let *