# Finite Switchboard State Machines Based on Cubic Sets

Authors: Naveed Yaqoob; Nabilah Abughazalah
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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 p-ideal (resp., a-ideal, q-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 mapλ:X→[0,1] is called a fuzzy set of X.”Definition 2 (see [1]).
“An interval valued fuzzy set (briefly, IVF-set)δ~A on X is defined as (1)δ~A=x,δA-x,δA+x:x∈X,where δA-(x)≤δA+(x), for all x∈X. Then the ordinary fuzzy sets δA-:X→[0,1] and δA+:X→[0,1] are called a lower fuzzy set and an upper fuzzy set of δ~, respectively. Let δ~A(x)=[δA-(x),δA+(x)]. Then (2)A=x,δ~Ax:x∈X,where δ~A:X→D[0,1].”Definition 3 (see [15]).
“LetX be a non-empty set. By a cubic set in X we mean a structure (3)φ=x,δ~φx,ωφx∣x∈Xin which δ~φ is an IVF set in X and ωφ is a fuzzy set in X.”Definition 4 (see [3]).
“A cubic finite state machine (cubic fsm, shortly) is a tripleT=〈P,X,φ〉, where P and X are finite non-empty sets, called the set of states and the set of input symbols, respectively, and φ=〈δ~φ,ωφ〉 is a cubic set in P×X×P.”“LetX∗ denote the set of all words of elements of X of finite length. Let ξ denote the empty word in X∗ and ξ denote the length of ξ for every ξ∈X∗. See [3].”Definition 5 (see [3]).
“LetT=〈P,X,φ〉 be a cubic fsm. Define a cubic set φ∗=〈δ~φ∗,ωφ∗〉 in P×X∗×P by (4)δ~φ∗p1,ξ,p2=1,1ifp1=p20,0ifp1≠p2,ωφ∗p1,ξ,p2=0ifp1=p21ifp1≠p2and(5)δ~φ∗p1,xa,p2=⋁p∈Prmin⁡δ~φ∗p1,x,p,δ~φp,a,p2ωφ∗p1,xa,p2=⋀p∈Pmax⁡ωφ∗p1,x,p,ωφp,a,p2for all p1,p2∈P, x∈X∗ and a∈X.”Lemma 6 (see [3]).
“LetT=〈P,X,φ〉 be a cubic fsm. Then (6)δ~φ∗p1,xy,p2=⋁p∈Prmin⁡δ~φ∗p1,x,p,δ~φ∗p,y,p2ωφ∗p1,xy,p2=⋀p∈Pmax⁡ωφ∗p1,x,p,ωφp,y,p2for all p1,p2∈P and x,y∈X∗."


## 3. Cubic Finite Switchboard State Machines

Definition 7.
A cubic fsmT=〈P,X,φ〉 is said to be switching if it satisfies(7)δ~φp1,x,p2=δ~φp2,x,p1,ωφp1,x,p2=ωφp2,x,p1for all p1,p2∈P and x∈X.Example 8.
LetP={p1,p2} and X={a,b}. Let φ=〈δ~φ,ωφ〉 be a cubic subset in P×X×P. Then the cubic fsm T=〈P,X,φ〉 is switching, as shown in Figure 1.Figure 1
A switching cubic fsmT=〈P,X,φ〉.Definition 9.
A cubic fsmT=〈P,X,φ〉 is said to be commutative if it satisfies(8)δ~φp1,xy,p2=δ~φp1,yx,p2,ωφp1,xy,p2=ωφp1,yx,p2for all p1,p2∈P and x,y∈X.Example 10.
LetP={p1,p2,p3} and X={a,b}. Let φ=〈δ~φ,ωφ〉 be a cubic subset in P×X×P. Then the cubic fsm T=〈P,X,φ〉 is commutative, as shown in Figure 2.Figure 2
A commutative cubic fsmT=〈P,X,φ〉.The cubic fsm shown in Figure1 is switching but not commutative because(9)0.2,0.6=δ~φp1,ab,p2≠δ~φp1,ba,p2=0.4,0.7,0.4=ωφp1,ab,p2≠ωφp1,ba,p2=0.6.Also, the cubic fsm shown in Figure 2 is commutative but not switching because(10)δ~φp1,a,p2≠δ~φp2,a,p1,ωφp1,a,p2≠ωφp2,a,p1.Proposition 11.
IfT=〈P,X,φ〉 is a commutative cubic fsm, then (11)δ~φ∗p1,xy,p2=δ~φ∗p1,yx,p2,ωφ∗p1,xy,p2=ωφ∗p1,yx,p2,for all p1,p2∈P, y∈X and x∈X∗.Proof.
Letp1,p2∈P, y∈X, and x∈X∗. We prove the result by induction on |x|=n. If n=0, then x=ξ; hence (12)δ~φ∗p1,xy,p2=δ~φ∗p1,ξy,p2=δ~φ∗p1,y,p2=δ~φ∗p1,yξ,p2=δ~φ∗p1,yx,p2,and(13)ωφ∗p1,xy,p2=ωφ∗p1,ξy,p2=ωφ∗p1,y,p2=ωφ∗p1,yξ,p2=ωφ∗p1,yx,p2.Therefore the result is true for n=0. Suppose that the result is true for u=n-1. That is, for all u∈X∗ with u=n-1, n>0. Let d∈X be such that x=ud. Then (14)δ~φ∗p1,xy,p2=δ~φ∗p1,udy,p2=⋁p∈Prmin⁡δ~φ∗p1,u,p,δ~φ∗p,dy,p2=⋁p∈Prmin⁡δ~φ∗p1,u,p,δ~φ∗p,yd,p2=δ~φ∗p1,uyd,p2=⋁p∈Prmin⁡δ~φ∗p1,uy,p,δ~φ∗p,d,p2=⋁p∈Prmin⁡δ~φ∗p1,yu,p,δ~φ∗p,d,p2=δ~φ∗p1,yud,p2=δ~φ∗p1,yx,p2,and(15)ωφ∗p1,xy,p2=ωφ∗p1,udy,p2=⋀p∈Pmax⁡ωφ∗p1,u,p,ωφ∗p,dy,p2=⋀p∈Pmax⁡ωφ∗p1,u,p,ωφ∗p,yd,p2=ωφ∗p1,uyd,p2=⋀p∈Pmax⁡ωφ∗p1,uy,p,ωφ∗p,d,p2=⋀p∈Pmax⁡ωφ∗p1,yu,p,ωφ∗p,d,p2=ωφ∗p1,yud,p2=ωφ∗p1,yx,p2.Hence the result is true for u=n. This completes the proof.Definition 12.
A cubic fsmT=〈P,X,φ〉 is called a cubic finite switchboard state machine (cubic fssm, shortly) if it is switching and commutative.Proposition 13.
IfT=〈P,X,φ〉 is a cubic fssm, then (16)δ~φ∗p1,x,p2=δ~φ∗p2,x,p1,ωφ∗p1,x,p2=ωφ∗p2,x,p1for all p1,p2∈P and x∈X∗.Proof.
Letp1,p2∈P and x∈X∗. We prove the result by induction on x=n. If n=0, then x=ξ, hence (17)δ~φ∗p1,x,p2=δ~φ∗p1,ξ,p2=δ~φ∗p2,ξ,p1=δ~φ∗p2,x,p1,and(18)ωφ∗p1,x,p2=ωφ∗p1,ξ,p2=ωφ∗p2,ξ,p1=ωφ∗p2,x,p1.Therefore the result is true for n=0. Assume that the result is true for u=n-1. That is, for all u∈X∗ with u=n-1, n>0, we have (19)δ~φ∗p1,x,p2=δ~φ∗p2,x,p1,ωφ∗p1,x,p2=ωφ∗p2,x,p1.Let a∈X and u∈X∗ be such that x=ua. Then (20)δ~φ∗p1,x,p2=δ~φ∗p1,ua,p2=⋁p∈Prmin⁡δ~φ∗p1,u,p,δ~φp,a,p2=⋁p∈Prmin⁡δ~φ∗p,u,p1,δ~φp2,a,p=⋁p∈Prmin⁡δ~φ∗p,u,p1,δ~φ∗p2,a,p=⋁p∈Prmin⁡δ~φ∗p2,a,p,δ~φ∗p,u,p1=δ~φ∗p2,au,p1=δ~φ∗p2,ua,p1=δ~φ∗p2,x,p1,and(21)ωφ∗p1,x,p2=ωφ∗p1,ua,p2=⋀p∈Pmax⁡ωφ∗p1,u,p,ωφp,a,p2=⋀p∈Pmax⁡ωφ∗p,u,p1,ωφp2,a,p=⋀p∈Pmax⁡ωφ∗p,u,p1,ωφ∗p2,a,p=⋀p∈Pmax⁡ωφ∗p2,a,p,ωφ∗p,u,p1=ωφ∗p2,au,p1=ωφ∗p2,ua,p1=ωφ∗p2,x,p1.This shows that the result is true for u=n. This completes the proof.Proposition 14.
IfT=〈P,X,φ〉 is a cubic fssm, then (22)δ~φ∗p1,xy,p2=δ~φ∗p1,yx,p2,ωφ∗p1,xy,p2=ωφ∗p1,yx,p2,for all p1,p2∈P and x,y∈X∗.Proof.
Proof easily follows from Propositions11 and 13.Definition 15.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φ^=〈δ~φ^,ωφ^〉 of P. Then 〈P,φ^,X,φ〉 is said to be a subsystem of T if and only if (23)δ~φ^p2⪰rmin⁡δ~φ^p1,δ~φp1,x,p2,ωφ^p2≤max⁡ωφ^p1,ωφp1,x,p2,for all p1,p2∈P and x∈X.If〈P,φ^,X,φ〉 is a subsystem of T, then we write simply φ^ for 〈P,φ^,X,φ〉.Example 16.
LetP={p1,p2,p3} and X={a,b}. Let φ=〈δ~φ,ωφ〉 and φ^=〈δ~φ^,ωφ^〉 be a two cubic subsets in P×X×P and P, respectively. Then subsystem φ^=〈δ~φ^,ωφ^〉 of the cubic fsm T=〈P,X,φ〉 is shown in Figure 3.Figure 3
A commutative cubic fsmT=〈P,X,φ〉.Theorem 17.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φ^=〈δ~φ^,ωφ^〉 of P. Then 〈P,φ^,X,φ〉 is subsystem of T if and only if (24)δ~φ^p2⪰rmin⁡δ~φ^p1,δ~φ∗p1,x,p2,ωφ^p2≤max⁡ωφ^p1,ωφ∗p1,x,p2.for all p1,p2∈P and x∈X∗.Proof.
Supposeφ^ is a subsystem of T. Let p1,p2∈P and x∈X∗. We prove the result by induction on x=n. If n=0, then x=ξ. Now, if p1=p2 then (25)δ~φ^p2=rmin⁡δ~φ^p1,δ~φ∗p1,x,p2,ωφ^p2=max⁡ωφ^p1,ωφ∗p1,x,p2.Now if p1≠p2, then (26)rmin⁡δ~φ^p1,δ~φ∗p1,x,p2=0,0⪯δ~φ^p2,and(27)max⁡ωφ^p1,ωφ∗p1,x,p2=1≥ωφ^p2.Thus the result is true for n=0. Suppose the result is true for all y∈X∗ such that y=n-1, n>0. Let x=ya, y∈X∗, a∈X and y=n-1. Then (28)rmin⁡δ~φ^p1,δ~φ∗p1,x,p2=rmin⁡δ~φ^p1,δ~φ∗p1,ya,p2=rmin⁡δ~φ^p1,⋁p∈Prmin⁡δ~φ∗p1,y,p,δ~φp,a,p2=⋁p∈Prmin⁡δ~φ^p1,δ~φ∗p1,y,p,δ~φp,a,p2⪯⋁p∈Prmin⁡δ~φ^p,δ~φp,a,p2⪯δ~φ^p2,and(29)max⁡ωφ^p1,ωφ^∗p1,x,p2=max⁡ωφ^p1,ωφ∗p1,ya,p2=max⁡ωφ^p1,⋀p∈Pmax⁡ωφ∗p1,y,p,ωφp,a,p2=⋀p∈Pmax⁡ωφ^p1,ωφ∗p1,y,p,ωφp,a,p2≥⋀p∈Pmax⁡ωφ^p,ωφp,a,p2≥ωφ^p2.The converse is obvious.Definition 18.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φg^=〈δ~φg^,ωφg^〉 of P. Then 〈P,φg^,X,φ〉 is said to be a good subsystem of T if and only if (30)δ~φg^p2=rmin⁡δ~φg^p1,δ~φp1,x,p2,ωφg^p2=max⁡ωφg^p1,ωφp1,x,p2,for all p1,p2∈P and x∈X.Example 19.
The good subsystem〈P,φg^,X,φ〉 of a cubic fssm T=〈P,X,φ〉 is shown in Figure 4.Figure 4
Good subsystemφg^ of cubic fssm T.Theorem 20.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φg^=〈δ~φg^,ωφg^〉 of P. Then 〈P,φg^,X,φ〉 is good subsystem of T if and only if (31)δ~φg^p2=rmin⁡δ~φg^p1,δ~φ∗p1,x,p2,ωφg^p2=max⁡ωφg^p1,ωφ∗p1,x,p2,for all p1,p2∈P and x∈X∗.Proof.
The proof is similar to the proof of Theorem17.Definition 21.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φs^=〈δ~φs^,ωφs^〉 of P. Then 〈P,φs^,X,φ〉 is said to be a strong subsystem of T if and only if (32)δ~φp1,x,p2≻0,0,ωφp1,x,p2<1,then(33)δ~φs^p2⪰δ~φs^p1,ωφs^p2≤ωφs^p1,for all p1,p2∈P and x∈X.Example 22.
The strong subsystem〈P,φs^,X,φ〉 of a cubic fssm T=〈P,X,φ〉 is shown in Figure 5.Figure 5
Strong subsystemφs^ of cubic fssm T.Remark 23.
A strong subsystem may be or may be not a good subsystem.Theorem 24.
Consider a cubic fssmT=〈P,X,φ〉 and a cubic subset φs^=〈δ~φs^,ωφs^〉 of P. Then 〈P,φs^,X,φ〉 is strong subsystem of T if and only if (34)δ~φ∗p1,x,p2≻0,0,ωφ∗p1,x,p2<1,then(35)δ~φs^p2⪰δ~φs^p1,ωφs^p2≤ωφs^p1,for all p1,p2∈P and x∈X∗.Proof.
Suppose thatφs^ is a strong subsystem. We prove the result by induction on x=n. If n=0, then x=ξ. Now, if p1=p2 then δ~φ∗(p2,ξ,p2)=[1,1] and ωφ∗(p2,ξ,p2)=0. Thus δ~φs^(p2)=δ~φs^(p2) and ωφs^(p2)=ωφs^(p2). Now if p1≠p2, then δ~φ∗(p1,ξ,p2)=[0,0] and ωφ∗(p1,ξ,p2)=1. Thus the result is true for n=0. Suppose the result is true for all y∈X∗ such that y=n-1, n>0. Let x=ya, y∈X∗, a∈X and y=n-1. Suppose that δ~φ∗(p1,x,p2)≻[0,0]. Then (36)δ~φ∗p1,x,p2≻0,0δ~φ∗p1,ya,p2≻0,0⋁p∈Prmin⁡δ~φ∗p1,y,p,δ~φ∗p,a,p2≻0,0.Thus there exists p∈P such that δ~φ∗(p1,y,p)≻[0,0] and δ~φ∗(p,a,p2)≻[0,0]. Hence δ~φs^(p2)⪰δ~φs^(p) and δ~φs^(p)⪰δ~φs^(p1). Thus δ~φs^(p2)⪰δ~φs^(p1). Now suppose that ωφ∗(p1,x,p2)<1. Then (37)ωφ∗p1,x,p2<1ωφ∗p1,ya,p2<1⋀p∈Pmax⁡ωφ∗p1,y,p,ωφ∗p,a,p2<1.Thus there exists p∈P such that ωφ∗(p1,y,p)<1 and ωφ∗(p,a,p2)<1. Hence ωφs^(p2)≤ωφs^(p) and ωφs^(p)≤ωφs^(p1). Thus ωφs^(p2)≤ωφs^(p1). The converse is obvious.Theorem 25.
LetT=〈P,X,φ〉 be a cubic fssm. Let φ1^ and φ2^ be two subsystems (resp., good subsystems, strong subsystems) of T. Then the following conditions hold.
(i)φ1^∧φ2^ is a subsystem (resp., good subsystem, strong subsystem) of T.
(ii)φ1^∨φ2^ is a subsystem (resp., good subsystem, strong subsystem) of T.Proof.
Suppose thatφ1^ and φ2^ be two subsystems of T. Then (38)δ~φ1^p2⪰rmin⁡δ~φ1^p1,δ~φp1,x,p2,ωφ1^p2≤max⁡ωφ1^p1,ωφp1,x,p2,also(39)δ~φ2^p2⪰rmin⁡δ~φ2^p1,δ~φp1,x,p2,ωφ2^p2≤max⁡ωφ2^p1,ωφp1,x,p2,for all p1,p2∈P and x∈X.
(i) Now to prove thatφ1^∧φ2^ is a subsystem of T. It is enough to prove that (40)δ~φ1^∧δ~φ2^p2⪰rmin⁡δ~φ1^∧δ~φ2^p1,δ~φp1,x,p2,and(41)ωφ1^∧ωφ2^p2≤max⁡ωφ1^∧ωφ2^p1,ωφp1,x,p2.Now(42)δ~φ1^∧δ~φ2^p2=rmin⁡δ~φ1^p2,δ~φ2^p2⪰rmin⁡rmin⁡δ~φ1^p1,δ~φp1,x,p2,rmin⁡δ~φ2^p1,δ~φp1,x,p2=rmin⁡rmin⁡δ~φ1^p1,δ~φ2^p1,δ~φp1,x,p2=rmin⁡δ~φ1^∧δ~φ2^p1,δ~φp1,x,p2,and (43)ωφ1^∧ωφ2^p2=max⁡ωφ1^p2,ωφ2^p2≤max⁡max⁡ωφ1^p1,ωφp1,x,p2,max⁡ωφ2^p1,ωφp1,x,p2=max⁡max⁡ωφ1^p1,ωφ2^p1,ωφp1,x,p2=max⁡ωφ1^∧ωφ2^p1,ωφp1,x,p2.Hence this shows that φ1^∧φ2^ is a subsystem of T.
(ii) Now to prove thatφ1^∨φ2^ is a subsystem of T. It is enough to prove that (44)δ~φ1^∨δ~φ2^p2⪰rmin⁡δ~φ1^∨δ~φ2^p1,δ~φp1,x,p2,and(45)ωφ1^∨ωφ2^p2≤max⁡ωφ1^∨ωφ2^p1,ωφp1,x,p2.Now(46)δ~φ1^∨δ~φ2^p2=rmax⁡(δ~φ1^p2,δ~φ2^p2⪰rmax⁡rmin⁡δ~φ1^p1,δ~φp1,x,p2,rmin⁡δ~φ2^p1,δ~φp1,x,p2=rmin⁡rmax⁡δ~φ1^p1,δ~φ2^p1,δ~φp1,x,p2=rmin⁡δ~φ1^∨δ~φ2^p1,δ~φp1,x,p2,and(47)ωφ1^∨ωφ2^p2=min⁡ωφ1^p2,ωφ2^p2≤min⁡max⁡ωφ1^p1,ωφp1,x,p2,max⁡ωφ2^p1,ωφp1,x,p2=max⁡min⁡ωφ1^p1,ωφ2^p1,ωφp1,x,p2=max⁡ωφ1^∨ωφ2^p1,ωφp1,x,p2.Hence this shows that φ1^∨φ2^ is a subsystem of T. 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.
LetT=〈P,X,φ〉 and V=〈Q,Y,ψ〉 be two cubic fssms and let X∩Y=∅. The Cartesian composition of T and V is denoted by(48)T∘V=P×Q,X∪Y,φ∘ψand is defined as follows:
(i)(49)δ~φ∘δ~ψp1,q,a,p2,q=δ~φp1,a,p2ωφ∘ωψp1,q,a,p2,q=ωφp1,a,p2for all p1,p2∈P, q∈Q and a∈X∪Y.
(ii)(50)δ~φ×δ~ψp,q1,a,p,q2=δ~ψq1,a,q2ωφ×ωψp,q1,a,p,q2=ωψq1,a,q2for all p∈P, q1,q2∈Q and a∈X∪Y.Proposition 27.
The “Cartesian composition” of two cubic fssms is a cubic fssm.Proof.
It is straightforward.Definition 28.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be the subsystems of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively, and let X∩Y=∅. The “Cartesian composition” of φ^ and ψ^ is denoted by (51)φ^∘ψ^=P×Q,φ^∘ψ^,X∪Y,φ∘ψand is defined as follows:
(i)(52)δ~φ^∘δ~ψ^p,q=rmin⁡δ~φ^p,δ~ψ^qωφ^∘ωψ^p,q=max⁡ωφ^p,ωψ^qfor all (p,q)∈P×Q.
(ii)(53)δ~φ∘δ~ψp1,q,a,p2,q=δ~φp1,a,p2ωφ∘ωψp1,q,a,p2,q=ωφp1,a,p2for all p1,p2∈P, q∈Q and a∈X∪Y.
(iii)(54)δ~φ∘δ~ψp,q1,a,p,q2=δ~ψq1,a,q2ωφ∘ωψp,q1,a,p,q2=ωψq1,a,q2for all p∈P, q1,q2∈Q and a∈X∪Y.Proposition 29.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be two subsystems (resp., good subsystems, strong subsystems) of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively. Then φ^∘ψ^ is a subsystem (resp., good subsystem, strong subsystem) of cubic fssm T∘V.Proof.
It is straightforward.Theorem 30.
Ifφ^∘ψ^ is a good subsystem of cubic fssm T∘V, then at least φ^ or ψ^ must be good subsystem.Proof.
Suppose thatφ^ and ψ^ are not good subsystems. Then there exist p1,p2∈P, q1,q2∈Q, x∈X, and y∈Y such that (55)δ~φ^p2≻rmin⁡δ~φ^p1,δ~φp1,x,p2,δ~ψ^q2≻rmin⁡δ~ψ^q1,δ~ψq1,y,q2also(56)ωφ^p2<max⁡ωφ^p1,ωφp1,x,p2,ωψ^q2<max⁡ωψ^q1,ωψq1,y,q2.Now by the definition (57)δ~φ∘δ~ψp,q1,y,p,q2=δ~ψq1,y,q2and(58)ωφ∘ωψp,q1,y,p,q2=ωψq1,y,q2.Now(59)δ~φ^∘δ~ψ^p,q2=rmin⁡δ~φ^p,δ~ψ^q2≻rmin⁡δ~φ^p,rmin⁡δ~ψ^q1,δ~ψq1,y,q2=rmin⁡rmin⁡δ~φ^p,δ~ψ^q1,δ~ψq1,y,q2=rmin⁡δ~φ^∘δ~ψ^p,q1,δ~ψq1,y,q2=rmin⁡δ~φ^∘δ~ψ^p,q1,δ~φ∘δ~ψp,q1,y,p,q2,and(60)ωφ^∘ωψ^p,q2=max⁡ωφ^p,ωψ^q2<max⁡ωφ^p,max⁡ωψ^q1,ωψq1,y,q2=max⁡max⁡ωφ^p,ωψ^q1,ωψq1,y,q2=max⁡ωφ^∘ωψ^p,q1,ωψq1,y,q2=max⁡ωφ^∘ωψ^p,q1,ωφ∘ωψp,q1,y,p,q2.This shows that φ^∘ψ^ is not good subsystem, a contradiction. This completes the proof.The following examples show that ifφ^=〈δ~φ^,ωφ^〉 is a good subsystem of T=〈P,X,φ〉 and ψ^=〈δ~ψ^,ωψ^〉 is not a good subsystem of V=〈Q,Y,ψ〉, then φ^∘ψ^ may or may not be a good subsystem.Example 31.
Considerφ^ is a good subsystem and ψ^ is not a good subsystem as shown in the Figures 6 and 7.
The Cartesian composition ofφ^ and ψ^ is shown in Figure 8. It is clear from Figure 8 that φ^∘ψ^ is a good subsystem.Figure 6
φ ^ is a good subsystem.Figure 7
ψ ^ is not a good subsystem.Figure 8
φ ^ ∘ ψ ^ is a good subsystem.Example 32.
Considerφ^ is a good subsystem and ψ^ is not a good subsystem as shown in Figures 9 and 10.
The Cartesian composition ofφ^ and ψ^ is shown in Figure 11. It is clear from Figure 11 that φ^∘ψ^ is not a good subsystemFigure 9
φ ^ is a good subsystem.Figure 10
ψ ^ is not a good subsystem.Figure 11
φ ^ ∘ ψ ^ is not a good subsystem.Now we can state a new proposition without its proof.Proposition 33.
Letφ^=〈δ~φ^,ωφ^〉 be a good subsystem of T=〈P,X,φ〉 and ψ^=〈δ~ψ^,ωψ^〉 be any subsystem of V=〈Q,Y,ψ〉; then φ^∘ψ^ is a good subsystem if and only if (61)δ~φp1,x,p2=δ~ψq1,y,q2,ωφp1,x,p2=ωψq1,y,q2,also(62)δ~φ^p1⪯δ~ψ^q1,ωφ^p1≥ωψ^q1,for all p1,p2∈P, q1,q2∈Q, x∈X and y∈Y.Theorem 34.
Ifφ^∘ψ^ is a strong subsystem of cubic fssm T∘V, then at least φ^ or ψ^ must be strong subsystem.Proof.
Suppose thatφ^ and ψ^ are not strong subsystems. Then there exist p1,p2∈P, q1,q2∈Q, x∈X and y∈Y such that (63)δ~φ^p2≺δ~φ^p1,δ~ψ^q2≺δ~ψ^q1also(64)ωφ^p2>ωφ^p1,ωψ^q2>ωψ^q1.We know that if δ~ψ(q1,y,q2)≻[0,0], then (δ~φ∘δ~ψ)((p,q1),y,(p,q2))≻[0,0]; also if ωψ(q1,y,q2)<1, then (ωφ∘ωψ)((p,q1),y,(p,q2))<1. Now consider that (65)δ~φ^∘δ~ψ^p,q2=rmin⁡δ~φ^p,δ~ψ^q2≺rmin⁡δ~φ^p,δ~ψ^q1=δ~φ^∘δ~ψ^p,q1and(66)ωφ^∘ωψ^p,q2=max⁡ωφ^p,ωψ^q2>max⁡ωφ^p,ωψ^q1=ωφ^∘ωψ^p,q1.This shows that φ^∘ψ^ is not a strong subsystem, a contradiction. This completes the proof.We can easily show with the help of an example that ifφ^=〈δ~φ^,ωφ^〉 is a strong subsystem of T=〈P,X,φ〉 and ψ^=〈δ~ψ^,ωψ^〉 is not a strong subsystem of V=〈Q,Y,ψ〉, then φ^∘ψ^ may or may not be a strong subsystem.Definition 35.
LetT=〈P,X,φ〉 and V=〈Q,Y,ψ〉 be two cubic fssms and let X∩Y=∅. The direct product of T and V is denoted by (67)T×V=P×Q,X×Y,φ×ψand is defined as follows:
(i)(68)δ~φ×δ~ψp1,q,x,p2,q=δ~φp1,x,p2ωφ×ωψp1,q,x,p2,q=ωφp1,x,p2for all p1,p2∈P, q∈Q and x∈X.
(ii)(69)δ~φ×δ~ψp,q1,y,p,q2=δ~ψq1,y,q2ωφ×ωψp,q1,y,p,q2=ωψq1,y,q2for all p∈P, q1,q2∈Q and y∈Y.
(iii)(70)δ~φ×δ~ψp1,q1,x,y,p2,q2=rmin⁡δ~φp1,x,p2,δ~ψq1,y,q2ωφ×ωψp1,q1,x,y,p2,q2=max⁡ωφp1,x,p2,ωψq1,y,q2for all (p1,q1) and (p2,q2)∈P×Q and (x,y)∈X×Y.Proposition 36.
The “direct product” of two cubic fssms is a cubic fssm.Proof.
It isstraightforward.Definition 37.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be the cubic subsystems of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively, and let X∩Y=∅. The “direct product” of φ^ and ψ^ is denoted by (71)φ^×ψ^=P×Q,φ^×ψ^,X×Y,φ×ψand is defined as follows:
(i)(72)δ~φ^×δ~ψ^p1,q1=rmin⁡δ~φ^p1,δ~ψ^q1ωφ^×ωψ^p1,q1=max⁡ωφ^p1,ωψ^q1for all (p1,q1)∈P×Q.
(ii)(73)δ~φ×δ~ψp1,q,x,p2,q=δ~φp1,x,p2ωφ×ωψp1,q,x,p2,q=ωφp1,x,p2for all p1,p2∈P, q∈Q and x∈X.
(iii)(74)δ~φ×δ~ψp,q1,y,p,q2=δ~ψq1,y,q2ωφ×ωψp,q1,y,p,q2=ωψq1,y,q2for all p∈P, q1,q2∈Q and y∈Y.
(iv)(75)δ~φ×δ~ψp1,q1,x,y,p2,q2=rmin⁡δ~φp1,x,p2,δ~ψq1,y,q2ωφ×ωψp1,q1,x,y,p2,q2=max⁡ωφp1,x,p2,ωψq1,y,q2for all (p1,q1) and (p2,q2)∈P×Q and (x,y)∈X×Y.Proposition 38.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be two subsystems (resp., good subsystems, strong subsystems) of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively. Then φ^×ψ^ is a subsystem (resp., good subsystem, strong subsystem) of cubic fssm T×V.Proof.
Itis straightforward.Theorem 39.
Ifφ^×ψ^ is a good subsystem of cubic fssm T×V, then at least φ^ or ψ^ must be good subsystem.Proof.
Similar to Theorem30.Theorem 40.
Ifφ^×ψ^ is a strong subsystem of cubic fssm T×V, then at least φ^ or ψ^ must be strong subsystem.Proof.
It is similar to Theorem34.


## 5. Homomorphism Problems in Cubic fssm

Definition 41.
LetT=〈P,X,φ〉 and V=〈Q,Y,ψ〉 be two cubic fssms. A pair (Φ,Ψ) of mappings Φ:P→Q and Ψ:X→Y is called homomorphism, written as (Φ,Ψ):T→V, if it satisfies (76)δ~φp1,x,p2⪯δ~ψΦp1,Ψx,Φp2,ωφp1,x,p2≥ωψΦp1,Ψx,Φp2,for all p1,p2∈P and x∈X.Definition 42.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be two subsystems of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively. A pair (Φ,Ψ) of mappings Φ:P→Q and Ψ:X→Y is called homomorphism, written as (Φ,Ψ):φ^→ψ^, if it satisfies(77)δ~φ^p1⪯δ~ψ^Φp1,ωφ^p1≥ωψ^Φp1,also(78)δ~φp1,x,p2⪯δ~ψΦp1,Ψx,Φp2,ωφp1,x,p2≥ωψΦp1,Ψx,Φp2,for all p1,p2∈P and x∈X.Example 43.
Letφ^=〈δ~φ^,ωφ^〉 and ψ^=〈δ~ψ^,ωψ^〉 be two subsystems of cubic fssms T=〈P,X,φ〉 and V=〈Q,Y,ψ〉, respectively, as shown in Figures 12 and 13.
We define(Φ,Ψ):φ^→ψ^ by (79)Φ:P→Q∣p→Φp=q1ifp∈p1,p2q2ifp∈p3,p4,Ψ:X→Y∣x→Ψx=aifx=bbifx=aThen by the routine calculation, we can see that (Φ,Ψ) is a homomorphism from φ^ to ψ^.Figure 12
A subsystemφ^ of cubic fssm T.Figure 13
A subsystemψ^ of cubic fssm V.Definition 44.
LetT=〈P,X,φ〉 and V=〈Q,Y,ψ〉 be two cubic fssms. A pair (Φ,Ψ) of mappings Φ:P→Q and Ψ:X→Y is called strong homomorphism, written as (Φ,Ψ):T→V, if it satisfies(80)δ~ψΦp1,Ψx,Φp2=⋁δ~φp1,x,p∣p∈P,Φp=Φp1,ωψΦp1,Ψx,Φp2=⋀ωφp1,x,p∣p∈P,Φp=Φp1,for all p1,p2∈P and x∈X.IfX=Y and Ψ is the identity map, then we simply write Φ:T→Q and say that Φ is a homomorphism or strong homomorphism accordingly. If (Φ,Ψ) is a strong homorphism with Φ is one-one, then (81)δ~ψΦp1,Ψx,Φp2=δ~φp1,x,p2,ωψΦp1,Ψx,Φp2=ωφp1,x,p2for all p1,p2∈P and x∈X.Theorem 45.
LetT=(P,X,φ) and Q=(Q,Y,ψ) be two cubic fsms. Let (Φ,Ψ):T→Q be an onto strong homomorphism. If T is a cubic fssm, then so is Q.Proof.
LetT=(P,X,φ) be a cubic fssm. Let q1,q2∈Q. Then there are p1,p2∈P such that Φ(p1)=q1 and Φ(p2)=q2. Let y1,y2∈Y. Then there exists x1,x2∈X such that Ψ(x1)=y1 and Ψ(x2)=y2. Since T is commutative, we have (82)δ~ψ∗q1,y1y2,q2=δ~ψ∗Φp1,Ψx1Ψx2,Φp2=δ~ψ∗Φp1,Ψx1x2,Φp2=⋁δ~φ∗p1,x1x2,p∣p∈P,Φp=Φp2=⋁δ~φ∗p1,x2x1,p∣p∈P,Φp=Φp2=δ~ψ∗Φp1,Ψx2x1,Φp2=δ~ψ∗q1,y2y1,q2,and(83)ωψ∗q1,y1y2,q2=ωψ∗Φp1,Ψx1Ψx2,Φp2=ωψ∗Φp1,Ψx1x2,Φp2=⋀ωφ∗p1,x1x2,p∣p∈P,Φp=Φp2=⋀ωφ∗p1,x2x1,p∣p∈P,Φp=Φp2=ωψ∗Φp1,Ψx2x1,Φp2=ωψ∗Φp1,Ψx2Ψx1,Φp2=ωψ∗q1,y2y1,q2.Hence Q is a commutative cubic fsm. Similarly we can show that Q is switching. Hence Q=(Q,Y,ψ) is also a cubic fssm.Theorem 46.
LetT=(P,X,φ) and Q=(Q,Y,ψ) be two cubic fssms. Let (Φ,Ψ):T→Q be a homomorphism. Then (84)δ~φ∗p1,x,p2⪯δ~ψ∗Φp1,Ψ∗x,Φp2,ωφ∗p1,x,p2≥ωψ∗Φp1,Ψ∗x,Φp2,for all p1,p2∈P and x∈X∗.Proof.
Letp1,p2∈P and x∈X∗. We prove the result by induction on x=n. If n=0, then x=ξ and so Ψ∗(x)=Ψ∗(ξ)=ξ. If p1=p2, then (85)δ~φ∗p1,x,p2=δ~φ∗p1,ξ,p2=1,1=δ~ψ∗Φp1,ξ,Φp2=δ~ψ∗Φp1,Ψ∗x,Φp2,and(86)ωφ∗p1,x,p2=ωφ∗p1,ξ,p2=0,0=ωψ∗Φp1,ξ,Φp2=ωψ∗Φp1,Ψ∗x,Φp2.If p1≠p2, then (87)δ~φ∗p1,x,p2=δ~φ∗p1,ξ,p2=0,0⪯δ~ψ∗Φp1,Ψ∗x,Φp2,and(88)ωφ∗p1,x,p2=ωφ∗p1,ξ,p2=1,1≥ωψ∗Φp1,Ψ∗x,Φp2.Therefore the result is true for n=0. Let us assume that the result is true for all b∈X∗ such that b=n-1,n>0. Let x=bc, where b∈X∗,c∈X and b=n-1. Then (89)δ~φ∗p1,x,p2=δ~φ∗p1,bc,p2=⋁p∈Pδ~φ∗p1,b,p∧δ~φ∗p,c,p2⪯⋁p∈Pδ~ψ∗Φp1,Ψ∗b,Φp∧δ~ψ∗Φp,Ψc,Φp2⪯⋁p∘∈Pδ~ψ∗Φp1,Ψ∗b,p∘∧δ~ψ∗p∘,Ψc,Φp2=δ~ψ∗Φp1,Ψ∗bΨc,Φp2=δ~ψ∗Φp1,Ψ∗bc,Φp2=δ~ψ∗Φp1,Ψ∗x,Φp2,and(90)ωφ∗p1,x,p2=ωφ∗p1,bc,p2=⋀p∈Pωφ∗p1,b,p∨ωφ∗p,c,p2≥⋀p∈Pωψ∗Φp1,Ψ∗b,Φp∨ωψ∗Φp,Ψc,Φp2≥⋀p∘∈Pωψ∗Φp1,Ψ∗b,p∘∨ωψ∗p∘,Ψc,Φp2=ωψ∗Φp1,Ψ∗bΨc,Φp2=ωψ∗Φp1,Ψ∗bc,Φp2=ωψ∗Φp1,Ψ∗x,Φp2.This is the required proof.


## 6. Conclusion

A finite switchboard state machine is a specialized finite state machines. We applied the notion of cubic sets to switchboard state machines. We introduced the concepts of (resp., good, strong) subsystem of cubic fssms. We proved that the “Cartesian composition” of two (resp., good, strong) subsystems of cubic fssm is (resp., good, strong) subsystem of cubic fssms. Similarly, the product of two (resp., good, strong) subsystems of cubic fssm is (resp. good, strong) subsystem of cubic fssms. We defined the homomorphism between two of cubic fssms and proved some related results. We provided many examples on each case.The construction of “P-union, P-intersection, R-union, and R-intersection” of subsystems of cubic fssms is still open.



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