Cartesian Product of Interval Neutrosophic Automata

2021·0.25 MB·English

by  V. Karthikeyan, R. Karuppaiya;

Turkish Journal of Computer and Mathematics Education Vol.12 No.12 (2021), 3708-3712

Research Article
Cartesian Product Of Interval Neutrosophic Automata
V. Karthikeyan 𝟏 R. Karuppaiya 𝟐
1Department of Mathematics, Government College of Engineering,Dharmapuri, Tamil Nadu, India.
2Department of Mathematics, Annamalai University,Chidambaram, Tamil Nadu, India.

Article History: Received: 11 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published
online: 23 May 2021
Abstract
We introduce Cartesian product of interval neutrosophic automata and prove that Cartesian product of cyclic interval
neutrosophic automata is cyclic
Key words: Cyclic, Cartesian product.
AMS Mathematics subject classification: 03D05,20M35,18 B20,68Q45, 68Q70,94 A45
1 Introduction
The neutrosophic set was introduced by Florentin Smarandache in 1999 [6]. The neutrosophic set is the
generalization of classical sets, fuzzy set [11] and so on. The fuzzy set was introduced by Zadeh in 1965[11].
Bipolar fuzzy set, YinYang bipolar fuzzy set, NPN fuzzy set were introduced by W. R. Zhang in [8, 9, 10].
A neutrosophic set N is classified by a Truth membership T , Indeterminacy membership I , and Falsity
N N
membership F , where T , I , and F are real standard and non-standard subsets of] 0−, 1+[. Interval-valued
N N N N
neutrosophic sets was introduced by Wang etal.,[7]. The concept of interval neutrosophic finite state machine
was introduced by Tahir Mahmood [5]. Generalized products of directable fuzzy automata are discussed in [1].
Retrievability, subsystems, and strong subsystems of INA were introduced in the papers [2, 3, 4].
In this paper, we introduce Cartesian product of interval neutrosophic automata and prove that Cartesian
product of cyclic interval neutrosophic automata is cyclic.

2 Preliminaries
2.1 Neutrosophic Set [6]
Let 𝑈 be the universal set.. A neutrosophic set (NS) 𝑁 in 𝑈 is classified by a truth membership T , an
N
indeterminacy membership I and a falsity membership F , where T , I , and F are real standard or non-
N N N N N
standard subsets of] 0−,1+[. That is
𝑁 ={〈 𝑥,𝑇 (𝑥),𝐼 (𝑥),𝐹 (𝑥) 〉,𝑥 ∈𝑈, 𝑇 , 𝐼 , 𝐹 ∈ ] 0−,1+[ } and 0− ≤𝑠𝑢𝑝 𝑇 (𝑥)+
𝑁 𝑁 𝑁 𝑁 𝑁 𝑁 𝑁
𝑠𝑢𝑝𝐼 (𝑥)+𝑠𝑢𝑝𝐹 (𝑥) ≤ 3+. We need to take the interval [0,1] for instead of ] 0−,1+[.
𝑁 𝑁

.𝟐.𝟐 Definition [𝟕]
An interval neutrosophic set (𝐼𝑁𝑆 for short) is 𝑁 = {〈𝛼 (𝑥),𝛽 (𝑥),𝛾 (𝑥)〉 |𝑥 ∈ 𝑈}
𝑁 𝑁 𝑁
= {〈𝑥,[𝑖𝑛𝑓 𝛼 (𝑥),𝑠𝑢𝑝 𝛼 (𝑥)],[𝑖𝑛𝑓 𝛽 (𝑥),𝑠𝑢𝑝 𝛽 (𝑥)],[𝑖𝑛𝑓 𝛾 (𝑥),𝑠𝑢𝑝 𝛾 (𝑥)]〉},𝑥 ∈ 𝑈, where 𝛼 (𝑥),
𝑁 𝑁 𝑁 𝑁 𝑁 𝑁 𝑁
𝛽 (𝑥), and 𝛾 (𝑥) representing the truth-membership, indeterminacy-membership and falsity membership for
𝑁 𝑁
each 𝑥 ∈ 𝑈. 𝛼 (𝑥), 𝛽 (𝑥), 𝛾 (𝑥) ⊆ [0,1] and the condition that 0 ≤ 𝑠𝑢𝑝 𝛼 (𝑥) + 𝑠𝑢𝑝 𝛽 (𝑥) +
𝑁 𝑁 𝑁 𝑁 𝑁
𝑠𝑢𝑝 𝛾 (𝑥) ≤ 3.
𝑁

𝟐.𝟑 Definition [𝟕]
An 𝐼𝑁𝑆 𝑁 is empty if 𝑖𝑛𝑓 𝛼 (𝑥) = 𝑠𝑢𝑝 𝛼 (𝑥) = 0,𝑖𝑛𝑓 𝛽 (𝑥) = 𝑠𝑢𝑝 𝛽 (𝑥) 1,𝑖𝑛𝑓 𝛾 (𝑥) = 𝑠𝑢𝑝 𝛾 (𝑥) =
𝑁 𝑁 𝑁 𝑁 𝑁 𝑁
1 for all 𝑥 ∈ 𝑈.
3 Interval Neutrosophic Automata
3.1 Definition [5]
𝑀 = (𝑄,𝛴,𝑁) is called interval neutrosophic automaton (𝐼𝑁𝐴 for short), where 𝑄 and 𝛴 are non-empty finite
sets called the set of states and input symbols respectively, and 𝑁 = {〈𝛼 (𝑥),𝛽 (𝑥),𝛾 (𝑥)〉} is an 𝐼𝑁𝑆 in 𝑄×
𝑁 𝑁 𝑁
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Research Article
𝛴×𝑄. The set of all words of finite length of 𝛴 is denoted by Σ∗. The empty word is denoted by 𝜖 and the length
of each 𝑥 ∈ Σ∗ is denoted by |𝑥|.
3.2 Definition [5]
𝑀 = (𝑄,𝛴,𝑁) be an 𝐼𝑁𝐴. Define an 𝐼𝑁𝑆 𝑁∗ = {〈𝛼𝑁∗(𝑥), 𝛽𝑁∗(𝑥),𝛾𝑁∗(𝑥)〉} in 𝑄×𝛴∗×𝑄 by
[1,1] 𝑖𝑓 𝑞 = 𝑞
𝑖 𝑗
𝛼𝑁∗(𝑞𝑖,𝜖,𝑞𝑗) = { [0,0] 𝑖𝑓 𝑞 ≠ 𝑞
𝑖 𝑗
[0,0] 𝑖𝑓 𝑞 = 𝑞
𝑖 𝑗
𝛽𝑁∗(𝑞𝑖,𝜖,𝑞𝑗) = { [1,1] 𝑖𝑓 𝑞 ≠ 𝑞
𝑖 𝑗
[0,0] 𝑖𝑓 𝑞 = 𝑞
𝑖 𝑗
𝛾𝑁∗(𝑞𝑖,𝜖,𝑞𝑗) = { [1,1] 𝑖𝑓 𝑞 ≠ 𝑞
𝑖 𝑗
𝛼𝑁∗(𝑞𝑖,𝑤,𝑞𝑗) = 𝛼𝑁∗(𝑞𝑖, 𝑥𝑦, 𝑞𝑗) = ∨𝑞𝑟 ∈𝑄[ 𝛼𝑁∗(𝑞𝑖, 𝑥, 𝑞𝑟)∧ 𝛼𝑁∗(𝑞𝑟, 𝑦, 𝑞𝑗)]
𝛽𝑁∗(𝑞𝑖,𝑤,𝑞𝑗) = 𝛽𝑁∗(𝑞𝑖, 𝑥𝑦, 𝑞𝑗) = ∧𝑞𝑟 ∈𝑄[ 𝛽𝑁∗(𝑞𝑖, 𝑥, 𝑞𝑟)∨ 𝛽𝑁∗(𝑞𝑟, 𝑦, 𝑞𝑗)]
𝛾𝑁∗(𝑞𝑖,𝑤,𝑞𝑗) = 𝛾𝑁∗(𝑞𝑖, 𝑥𝑦, 𝑞𝑗) = ∧𝑞𝑟 ∈𝑄[ 𝛾𝑁∗(𝑞𝑖, 𝑥, 𝑞𝑟)∨ 𝛾𝑁∗(𝑞𝑟, 𝑦, 𝑞𝑗)] ∀ 𝑞𝑖, 𝑞𝑗 ∈𝑄,𝑤 =𝑥𝑦 ,𝑥 ∈
Σ∗ and 𝑦 ∈Σ.
4 Cartesian Composition of Interval Neutrosophic Automata
4.1 Definition
Let 𝑀 =(𝑄, Σ ,𝑁),𝑖 =1,2 be interval neutrosophic automata and let Σ ∩ Σ =∅. Let 𝑀 × 𝑀 =
𝑖 𝑖 𝑖 𝑖 1 2 1 2
(𝑄 × 𝑄 , Σ ∪ Σ , 𝑁 × 𝑁 ), where
1 2 1 2 1 2
𝛼 (𝑞,𝑎,𝑞 )>[0,0] 𝑖𝑓 𝑎 ∈ Σ 𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
(α1×α2)((𝑞𝑖,𝑞𝑗),𝑎,(𝑞𝑘,𝑞𝑙))= {𝛼2(𝑞𝑖,𝑎,𝑞𝑘)>[0,0] 𝑖𝑓 𝑎 ∈ Σ2 𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛽 (𝑞,𝑎,𝑞 )<[1,1] 𝑖𝑓 𝑎 ∈ Σ 𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
(𝛽1×𝛽2)((𝑞𝑖,𝑞𝑗),𝑎,(𝑞𝑘,𝑞𝑙))= {𝛽2(𝑞𝑖,𝑎,𝑞𝑘)<[1,1] 𝑖𝑓 𝑎 ∈ Σ2 𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛾 (𝑞 ,𝑎,𝑞 )<[1,1] 𝑖𝑓 𝑎 ∈ Σ 𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
( 𝛾1× 𝛾2)((𝑞𝑖,𝑞𝑗),𝑎,(𝑞𝑘,𝑞𝑙))= { 𝛾2(𝑞𝑖,𝑎,𝑞𝑘)<[1,1] 𝑖𝑓 𝑎 ∈ Σ2 𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
∀(𝑞,𝑞 ), (𝑞 ,𝑞) ∈ 𝑄 ×𝑄 , 𝑎 ∈ Σ ∪ Σ . Then 𝑀 × 𝑀 is called the Cartesian product of interval
𝑖 𝑗 𝑘 𝑙 1 2 1 2 1 2
neutrosophic automata.
4.2 Definition
Let 𝑀 = (𝑄,𝛴,𝑁) be an INA. 𝑀 is cyclic if ∃ 𝑞 ∈ Q such that Q = S(𝑞).
𝑖 𝑖
4.3 Definition [2]
Let 𝑀 = (𝑄,𝛴,𝑁) be INA. M is connected if ∀ 𝑞 ,𝑞 and ∃ a ∈ Σ such that either α (𝑞,𝑎,𝑞 )>[0,0],
𝑗 𝑖 N 𝑖 𝑗
β (𝑞,𝑎,𝑞 )<[1,1], 𝛾 (𝑞,𝑎,𝑞 )<[1,1] or
N 𝑖 𝑗 𝑁 𝑖 𝑗
α (𝑞 ,𝑎,𝑞)>[0,0], β (𝑞 ,𝑎,𝑞 )<[1,1], 𝛾 (𝑞 ,𝑎,𝑞)<[1,1].
N 𝑗 𝑖 N 𝑗 𝑖 𝑁 𝑗 𝑖

4.4 Definition [2]

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Turkish Journal of Computer and Mathematics Education Vol.12 No.12 (2021), 3708-3712

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Let 𝑀 = (𝑄,𝛴,𝑁) be INA. M is strongly connected if for every 𝑞,𝑞 ∈ 𝑄, there exists u ∈ Σ∗ such that
𝑖 𝑗
α𝑁∗(𝑞𝑖,𝑢,𝑞𝑗)>[0,0],β𝑁∗(𝑞𝑖,𝑢,𝑞𝑗)<[1,1],γ𝑁∗(𝑞𝑖,𝑢,𝑞𝑗)<[1,1].

Theorem 4.1 Let 𝑀 =(𝑄,Σ ,𝑁),𝑖 =1,2 be interval neutrosophic automata and let Σ ∩ Σ = ∅. Let
𝑖 𝑖 𝑖 𝑖 1 2
𝑀 × 𝑀 = (𝑄 ×𝑄 ,Σ ∪ Σ ,𝑁 ×𝑁 ) be the Cartesian product of 𝑀 and 𝑀 . Then ∀ 𝑥 ∈Σ ∗ ∪ Σ ∗,𝑥 ≠
1 2 1 2 1 2 1 2 1 2 1 2
𝜖
𝛼 (𝑞 ,𝑥,𝑞 )>[0,0] 𝑖𝑓 𝑥 ∈Σ ∗ 𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
(α1×α2)∗((𝑞𝑖,𝑞𝑗),𝑥,(𝑞𝑘,𝑞𝑙))= {𝛼2(𝑞𝑖,𝑥,𝑞𝑘)>[0,0] 𝑖𝑓 𝑥 ∈Σ2∗𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛽 (𝑞,𝑥,𝑞 )<[1,1] 𝑖𝑓 𝑥 ∈ Σ ∗ 𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
(𝛽1×𝛽2)∗((𝑞𝑖,𝑞𝑗),𝑥,(𝑞𝑘,𝑞𝑙))= {𝛽2(𝑞𝑖,𝑥,𝑞𝑘)<[1,1] 𝑖𝑓 𝑥 ∈ Σ2∗ 𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛾 (𝑞 ,𝑥,𝑞 )<[1,1] 𝑖𝑓 𝑥 ∈ Σ ∗𝑎𝑛𝑑 𝑞 = 𝑞
1 𝑖 𝑘 1 𝑗 𝑙
(𝛾1× 𝛾2)∗((𝑞𝑖,𝑞𝑗),𝑥,(𝑞𝑘,𝑞𝑙))= { 𝛾2(𝑞𝑖,𝑥,𝑞𝑘)<[1,1] 𝑖𝑓 𝑥 ∈ Σ1∗𝑎𝑛𝑑 𝑞𝑖 = 𝑞𝑘
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
∀(𝑞,𝑞 ), (𝑞 ,𝑞) ∈ 𝑄 ×𝑄 .
𝑖 𝑗 𝑘 𝑙 1 2
Proof . Let 𝑥 ∈Σ ∗ ∪ Σ ∗,𝑥 ≠ 𝜖 and let |𝑥|=𝑚. Let 𝑥 ∈Σ ∗. The result is trivial if 𝑚=1. Let the
1 2 1
result is true ∀ 𝑦 ∈ Σ ∗, |𝑦|=𝑚−1,𝑚 >1. Let 𝑥 =𝑎𝑦 where 𝑎 ∈ Σ ,𝑦 ∈ Σ ∗. Now,
1 1 1
(α ×α )∗((𝑞,𝑞 ),𝑥,(𝑞 ,𝑞))=(α ×α )∗((𝑞,𝑞 ),𝑎𝑦,(𝑞 ,𝑞))
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
= ∨ {(α ×α )((𝑞,𝑞 ),𝑎,(𝑞 ,𝑞 ))∧(α ×α )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞))}
(𝑞𝑟, 𝑞𝑠) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑟 𝑠 1 2 𝑟 𝑠 𝑘 𝑙
= ∨ {α (𝑞, 𝑎, 𝑞 ) ∧ (α ×α )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞))}
𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑁2 𝑟 𝑠 𝑘 𝑙
∨ {α (𝑞, 𝑎, 𝑞 ) ∧ α ∗(𝑞 ,𝑦,𝑞 )}>[0,0] 𝑖𝑓 𝑞 = 𝑞
= { 𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑟 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
α ∗(𝑞 ,𝑎𝑦,𝑞 )>[0,0] 𝑖𝑓 𝑞 = 𝑞
= { 𝑁1 𝑖 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(β ×β )∗((𝑞,𝑞 ),𝑥,(𝑞 ,𝑞))=(β ×β )∗((𝑞,𝑞 ),𝑎𝑦,(𝑞 ,𝑞))
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
= ∧ {(β ×β )((𝑞,𝑞 ),𝑎,(𝑞 ,𝑞 ))∨(β ×β )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞 ))}
(𝑞𝑟, 𝑞𝑠) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑟 𝑠 𝑁1 𝑁2 𝑟 𝑠 𝑘 𝑙
= ∧ {β (𝑞, 𝑎, 𝑞 ) ∨ (β ×β )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞))}
𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑁2 𝑟 𝑠 𝑘 𝑙
∧ {β (𝑞 , 𝑎, 𝑞 ) ∨ β ∗(𝑞 ,𝑦,𝑞 )}<[1,1] 𝑖𝑓 𝑞 = 𝑞
= { 𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑟 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
β ∗(𝑞,𝑎𝑦,𝑞 )<[1,1] 𝑖𝑓 𝑞 = 𝑞
= { 𝑁1 𝑖 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(γ ×γ )∗((𝑞,𝑞 ),𝑥,(𝑞 ,𝑞))=(γ ×γ )∗((𝑞 ,𝑞 ),𝑎𝑦,(𝑞 ,𝑞))
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
= ∧ {(γ ×γ )((𝑞,𝑞 ),𝑎,(𝑞 ,𝑞 ))∨(γ ×γ )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞 ))}
(𝑞𝑟, 𝑞𝑠) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑟 𝑠 𝑁1 𝑁2 𝑟 𝑠 𝑘 𝑙
= ∧ {γ (𝑞, 𝑎, 𝑞 ) ∨ (γ ×γ )∗((𝑞 ,𝑞 ),𝑦,(𝑞 ,𝑞))}
𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑁2 𝑟 𝑠 𝑘 𝑙
∧ {γ (𝑞, 𝑎, 𝑞 ) ∨ γ ∗(𝑞 ,𝑦,𝑞 )}<[1,1] 𝑖𝑓 𝑞 = 𝑞
= { 𝑞𝑟 ∈ 𝑄1 𝑁1 𝑖 𝑟 𝑁1 𝑟 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
γ ∗(𝑞,𝑎𝑦,𝑞 )<[1,1] 𝑖𝑓 𝑞 = 𝑞
= { 𝑁1 𝑖 𝑘 𝑗 𝑙
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
The result is follows by induction. The Proof is similar if 𝑦 ∈ Σ ∗.
2

Theorem 4.2 Let 𝑀 =(𝑄,Σ ,𝑁),𝑖 =1,2 be INA and let Σ ∩ Σ = ∅. Then ∀ 𝑥 ∈ Σ ∗,𝑦 ∈ Σ ∗,
𝑖 𝑖 𝑖 𝑖 1 2 1 2
(α ×α )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞 ,𝑞 ))= α ∗(𝑝,𝑥,𝑞) ∧ α ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
= (α ×α )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞,𝑞 ))
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(β ×β )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))= β ∗(𝑝,𝑥,𝑞) ∨ β ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
= (β ×β )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞,𝑞 ))
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(γ ×𝛾 )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))= γ ∗(𝑝,𝑥,𝑞) ∨ γ ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
= (γ ×γ )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞,𝑞 )),
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(𝑝,𝑝 ),(𝑞,𝑞 )∈𝑄 ×𝑄 .
𝑖 𝑗 𝑖 𝑗 1 2
Proof .
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Let ∈ Σ ∗,𝑦 ∈ Σ ∗, (𝑝, 𝑝 ),(𝑞 , 𝑞 )∈𝑄 ×𝑄 . If 𝑥 = 𝜖 =𝑦, then 𝑥𝑦=𝜖. Suppose (𝑝, 𝑝 )=
1 2 𝑖 𝑗 𝑖 𝑗 1 2 𝑖 𝑗
(𝑞 , 𝑞 ). Then 𝑝 = 𝑞 and 𝑝 =𝑞 . Hence
𝑖 𝑗 𝑖 𝑖 𝑗 𝑗
(α ×α )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[1,1]=[1,1] ∧[1,1]= α ∗(𝑝,𝑥,𝑞) ∧ α ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
(β ×β )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[0,0]=[0,0]∨[0,0]= β ∗(𝑝,𝑥,𝑞)∨β ∗(𝑝,𝑥,𝑞)
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁1 𝑖 𝑖
(γ ×𝛾 )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[0,0]=[0,0]∨[0,0]= γ ∗(𝑝,𝑥,𝑞)∨ γ ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
If (𝑝,𝑝 ) ≠ (𝑞,𝑞 ), then either 𝑝 ≠ 𝑞 or 𝑝 ≠𝑞 .
𝑖 𝑗 𝑖 𝑗 𝑖 𝑖 𝑗 𝑗
Thus, α ∗(𝑝,𝑥,𝑞) ∧ α ∗(𝑝 ,𝑦,𝑞 )=[0,0], β ∗(𝑝,𝑥,𝑞)∨β ∗(𝑝,𝑥,𝑞)=[1,1],
𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗 𝑁1 𝑖 𝑖 𝑁1 𝑖 𝑖
γ ∗(𝑝,𝑥,𝑞)∨ γ ∗(𝑝 ,𝑦,𝑞 )=[1,1].
𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
Hence (α ×α )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[0,0] = α ∗(𝑝,𝑥,𝑞) ∧ α ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
(β ×β )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[1,1]= β ∗(𝑝,𝑥,𝑞)∨β ∗(𝑝,𝑥,𝑞)
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁1 𝑖 𝑖
(γ ×𝛾 )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))=[1,1]= γ ∗(𝑝,𝑥,𝑞)∨ γ ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
If x = 𝜖 and y ≠ 𝜖 or x ≠ 𝜖 and y = 𝜖, then the result follows by Theorem 4.1. Suppose x ≠ 𝜖 and y ≠ 𝜖.
Now,
(α ×α )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞 ,𝑞 ))= ∨ {(α ×α )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑟))∧
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 (𝑟𝑖, 𝑟𝑗) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(α ×α )∗((𝑟,𝑟),𝑦,(𝑞,𝑞 ))}
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= ∨ {(α ×α )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑝 ))∧ (α ×α )∗((𝑟,𝑝 ),𝑦,(𝑞 ,𝑞 ))}
𝑟𝑖 ∈ 𝑄1 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= α ∗(𝑝,𝑥,𝑞 ) ∧ α ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
(β ×β )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))= ∧ {(β ×β )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑟))∨
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 (𝑟𝑖, 𝑟𝑗) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(β ×β )∗((𝑟,𝑟),𝑦,(𝑞,𝑞 ))}
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= ∧ {(β ×β )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑝 ))∨ (β ×𝛽 )∗((𝑟,𝑝 ),𝑦,(𝑞,𝑞 ))}
𝑟𝑖 ∈ 𝑄1 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= 𝛽 ∗(𝑝,𝑥,𝑞) ∨ 𝛽 ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
(γ ×𝛾 )∗((𝑝,𝑝 ),𝑥𝑦,(𝑞,𝑞 ))= ∧ {(γ ×γ )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑟))∨
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 (𝑟𝑖, 𝑟𝑗) ∈ 𝑄1× 𝑄2 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
(γ ×γ )∗((𝑟,𝑟),𝑦,(𝑞 ,𝑞 ))}
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= ∧ {(γ ×γ )∗((𝑝,𝑝 ),𝑥,(𝑟,𝑝 ))∨ (γ ×γ )∗((𝑟,𝑝 ),𝑦,(𝑞,𝑞 ))}
𝑟𝑖 ∈ 𝑄1 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗
= γ ∗(𝑝,𝑥,𝑞) ∨ γ ∗(𝑝 ,𝑦,𝑞 )
𝑁1 𝑖 𝑖 𝑁2 𝑗 𝑗
Similarly
(α ×α )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞 ,𝑞 ))= α ∗(𝑝 ,𝑦,𝑞 ) ∧ α ∗(𝑝,𝑥,𝑞 )
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁2 𝑗 𝑗 𝑁1 𝑖 𝑖
(β ×β )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞,𝑞 ))= 𝛽 ∗(𝑝 ,𝑦,𝑞 ) ∨ 𝛽 ∗(𝑝,𝑥,𝑞)
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁2 𝑗 𝑗 𝑁1 𝑖 𝑖
(γ ×𝛾 )∗((𝑝,𝑝 ),𝑦𝑥,(𝑞,𝑞 ))= γ ∗(𝑝 ,𝑦,𝑞 ) ∨ γ ∗(𝑝,𝑥,𝑞).
𝑁1 𝑁2 𝑖 𝑗 𝑖 𝑗 𝑁2 𝑗 𝑗 𝑁1 𝑖 𝑖
Theorem 4.3 Let 𝑀 =(𝑄,Σ ,𝑁),𝑖 =1,2 be INA and let Σ ∩ Σ = ∅. Cartesian product of 𝑀 ×𝑀 is
𝑖 𝑖 𝑖 𝑖 1 2 1 2
cyclic iff 𝑀 and 𝑀 are cyclic.
1 2
Proof. Let 𝑀 and 𝑀 are cyclic. Then 𝑄 =𝑆(𝑞) and 𝑄 =𝑆(𝑝 ) for some 𝑞 ∈𝑄 ,𝑝 ∈ 𝑄 . Let
1 2 1 𝑖 2 𝑗 𝑖 1 𝑗 2
(𝑞 ,𝑝) ∈ 𝑄 × 𝑄 . Then ∃ 𝑥 ∈ Σ ∗ and 𝑦 ∈ Σ ∗ such that
𝑘 𝑙 1 2 1 2
α ∗(𝑞,𝑥,𝑞 )>[0,0],β ∗(𝑞 ,𝑥,𝑞 )<[1,1],γ ∗(𝑞,𝑥,𝑞 ) < [1, 1] and
𝑁1 𝑖 𝑘 𝑁1 𝑖 𝑘 𝑁1 𝑖 𝑘
α ∗(𝑝 ,𝑦,𝑝)>[0,0],β ∗(𝑝 ,𝑦,𝑝)<[1,1],γ ∗(𝑝 ,𝑦,𝑝) < [1, 1]. Thus
𝑁2 𝑗 𝑙 𝑁2 𝑗 𝑙 𝑁2 𝑗 𝑙
(α ×α )∗((𝑞,𝑝 ),𝑥𝑦,(𝑞 ,𝑝))= α ∗(𝑞,𝑥,𝑞 ) ∧ α ∗(𝑝 ,𝑦,𝑝)>[0,0]
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙
(β ×β )∗((𝑞,𝑝 ),𝑥𝑦,(𝑞 ,𝑝))= β ∗(𝑞,𝑥,𝑞 ) ∨ β ∗(𝑝 ,𝑦,𝑝)<[1,1]
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙
(γ ×𝛾 )∗ ((𝑞,𝑝 ),𝑥𝑦,(𝑞 ,𝑝))= γ ∗(𝑞,𝑥,𝑞 ) ∨ γ ∗(𝑝 ,𝑦,𝑝)<[1,1].
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙
Hence (𝑞 ,𝑝) ∈𝑆((𝑞,𝑝 )). 𝑄 × 𝑄 = 𝑆((𝑞,𝑝 )). Hence 𝑀 ×𝑀 is cyclic.
𝑘 𝑙 𝑖 𝑗 1 2 𝑖 𝑗 1 2
Conversely, let 𝑀 ×𝑀 is cyclic. Then 𝑄 × 𝑄 = 𝑆((𝑞 ,𝑝 )) for some (𝑞,𝑝 )∈ 𝑄 × 𝑄 .
1 2 1 2 𝑖 𝑗 𝑖 𝑗 1 2
Let 𝑞 ∈ 𝑄 and 𝑝 ∈ 𝑄 . Then ∃ 𝑤 ∈ (Σ ∪ Σ )∗ such that
𝑘 1 𝑙 2 1 2
(α ×α )∗((𝑞 ,𝑝 ),𝑤,(𝑞 ,𝑝))>[0,0], (β ×β )∗((𝑞,𝑝 ),𝑤,(𝑞 ,𝑝))<[1,1] and
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
(γ ×𝛾 )∗ ((𝑞,𝑝 ),𝑤,(𝑞 ,𝑝)) < [1, 1]. Then by the theorem 4.2 ∃ 𝑢 ∈Σ ∗ and 𝑣 ∈ Σ ∗ such that
𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙 1 2
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Turkish Journal of Computer and Mathematics Education Vol.12 No.12 (2021), 3708-3712

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α ∗(𝑞,𝑢,𝑞 ) ∧ α ∗(𝑝 ,𝑣,𝑝)= (α ×α )∗((𝑞,𝑝 ),𝑤,(𝑞 ,𝑝)) >[0,0]
𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
β ∗(𝑞,𝑢,𝑞 ) ∨ β ∗(𝑝 ,𝑣,𝑝)= (β ×β )∗((𝑞,𝑝 ),𝑤,(𝑞 ,𝑝))<[1,1]
𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
γ ∗(𝑞,𝑢,𝑞 ) ∨ γ ∗(𝑝 ,𝑣,𝑝)= (γ ×𝛾 )∗ ((𝑞 ,𝑝 ),𝑤,(𝑞 ,𝑝)) < [1, 1].
𝑁1 𝑖 𝑘 𝑁2 𝑗 𝑙 𝑁1 𝑁2 𝑖 𝑗 𝑘 𝑙
Hence ∃ 𝑢 ∈Σ ∗ and 𝑣 ∈ Σ ∗ such that α ∗(𝑞,𝑢,𝑞 ) >[0,0], β ∗(𝑞,𝑢,𝑞 )<[1,1], γ ∗(𝑞,𝑢,𝑞 ) <[1,1]
1 2 𝑁1 𝑖 𝑘 𝑁1 𝑖 𝑘 𝑁1 𝑖 𝑘
and α ∗(𝑝 ,𝑣,𝑝)>[0,0], β ∗(𝑝 ,𝑣,𝑝)<[1,1],γ ∗(𝑝 ,𝑣,𝑝)<[1,1]. Thus 𝑞 ∈𝑆(𝑞 ) and 𝑝 ∈𝑆(𝑝 ).
𝑁2 𝑗 𝑙 𝑁2 𝑗 𝑙 𝑁2 𝑗 𝑙 𝑘 𝑖 𝑙 𝑗
Hence 𝑄 ∈𝑆(𝑞) and 𝑄 ∈𝑆(𝑝 ). Therefore 𝑀 ×𝑀 is cyclic.
1 𝑖 2 𝑗 1 2
5 Conclusion
The purpose of this paper is to study the Cartesian product of INA. We prove that Cartesian product of cyclic of
interval neutrosophic automata is cyclic.
References
[1] V. Karthikeyan, N. Mohanarao, and S. Sivamani, Generalized products of directable fuzzy Automata,
Material Today: Proceedings, 37(2), 2021, 35313533.
[2] V. Karthikeyan, and R. Karuppaiya, Retrievebility in Interval Neutrosophic Automata, Advance in
Mathematics: Scientific Journal, 9(4), 2020, 1637-1644.
[3] V. Karthikeyan, and R. Karuppaiya, Subsystems of Interval Neutrosophic Automata, Advance in
Mathematics: Scientific Journal, 9(4), 2020, 1653-1659.
[4] V. Karthikeyan, and R. Karuppaiya, Strong subsystems of Interval Neutrosophic Automata, Advance in
Mathematics: Scientific Journal, 9(4), 2020, 1645-1651.
[5] T. Mahmood, and Q. Khan, Interval neutrosophic finite switchboard state machine, Afr. Mat. 20(2), 2016,
191-210.
[6] F. Smarandache, A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, set and Logic,
Rehoboth: American Research Press, 1999.
[7] H. Wang, F. Smarandache, Y.Q. Zhang, and R. Sunderraman, Interval Neutrosophic Sets and Logic, Theory
and Applications in Computing, 5, 2005, Hexis, Phoenix, AZ.
[8] W. R. Zhang, Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and
multiagent decision analysis, Proc. 1st Int. Joint Conf. North American Fuzzy Information Processing Society
Biannual Conf., San Antonio, TX, USA, 1994, 305–309.
[9] W. R. Zhang, YinYang bipolar fuzzy sets, Proc. IEEE World Congr. Computational Intelligence, Anchorage,
Alaska, 1998, 835–840.
[10] W. R. Zhang, NPN Fuzzy Sets and NPN Qualitative-Algebra: A Computational Framework for Bipolar
Cognitive Modeling and Multiagent Decision Analysis, IEEE Trans. on Sys., Man, and Cybern. 26(8), 1996,
561-575.
[11] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3), 1965, 338-353.
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