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 117 
 

Neutrosophic Hedge Algebras 
 

Florentin Smarandache 
University of New Mexico 

705 Gurley Ave., Gallup, New Mexico 87301, USA 
Phone: +1 505-277-0111 
fsmarandache@gmail.com 

 
Abstract 
We introduce now for the first time the neutrosophic hedge algebras as an extension of 

classical hedge algebras, together with an application of neutrosophic hedge algebras. 
 
1. Introduction 
The classical hedge algebras deal with linguistic variables. In neutrosophic environment we 

have introduced the neutrosophic linguistic variables. We have defined neutrosophic partial 
relationships between single-valued neutrosophic numbers. Neutrosophic operations are used in 
order to aggregate the neutrosophic linguistic values. 

 
2. Materials and Methods  
We introduce now, for the first time, the Neutrosophic Hedge Algebras, as extension of 

classical Hedge Algebras. 
 

Let's consider a Linguistic Variable: 
with 𝐷𝑜𝑚(𝑥) as the word domain of 𝑥, whose each element is a word (label), or string of 

words. 
Let 𝒜 be an attribute that describes the value of each element 𝑥 ∈ 𝐷𝑜𝑚(𝑥), as follows: 
𝒜: 𝐷𝑜𝑚(𝑥) → [0, 1] .    (1) 
𝒜(𝑥) is the neutrosophic value of 𝑥 with respect to this attribute: 
𝐴(𝑥) = 〈𝑡 , 𝑖 , 𝑓 〉,      (2) 
where 𝑡 , 𝑖 , 𝑓 ∈ [0, 1], such that 
 
– 𝑡  means the degree of value of 𝑥; 
– 𝑖  means the indeterminate degree of value of 𝑥; 
– 𝑓  means the degree of non-value of 𝑥. 

 
We may also use the notation: 𝑥〈𝑡 , 𝑖 , 𝑓 〉. 
A neutrosophic partial relationship ≤  on 𝐷𝑜𝑚(𝑥), defined as follows: 
𝑥〈𝑡 , 𝑖 , 𝑓 〉 ≤ 𝑦〈𝑡 , 𝑖 , 𝑓 〉,    (3) 
if and only if 𝑡 ≤ 𝑡 , and 𝑖 ≥ 𝑖 , 𝑓 ≥ 𝑓 . 
Therefore, (𝐷𝑜𝑚(𝑥), ≤ ) becomes a neutros-ophic partial order set (or neutrosophic poset), 

and ≤  is called a neutrosophic inequality. 
Let 𝐶 = {0, 𝑤, 1} be a set of constants, 𝐶 ⊂ 𝐷𝑜𝑚(𝑥), where: 

 
– 0 = the least element, or 0〈 , , 〉; 
– w = the neutral (middle) element, or 𝑤〈 . , . , . 〉; 
– and 1 = the greatest element, or 1〈 , , 〉. 

 
Let 𝐺 be a word-set of two neutrosophic generators, 𝐺 ⊂ 𝐷𝑜𝑚(𝑥), qualitatively a negative 

primary neutrosophic term (denoted 𝑔 ), and the other one that is qualitatively a positive primary 
neutrosophic term (denoted 𝑔 ), such that: 



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0 ≤ 𝑔 ≤ 𝑤 ≤ 𝑔 ≤ 1,   (4) 
or transcribed using the neutrosophic com-ponents: 

0〈 , , 〉 ≤ 𝑔 〈 , , 〉 ≤ 𝑤〈 . , . , . 〉 

≤ 𝑔
〈 , , 〉

≤ 1〈 , , 〉, 

where 
 

– 0 ≤ 𝑡 ≤ 0.5 ≤ 𝑡 ≤ 1 (here there are classical inequalities) 
– 1 ≥ 𝑖 ≥ 0.5 ≥ 𝑖 ≥ 0, and 
– 1 ≥ 𝑓 ≥ 0.5 ≥ 𝑓 ≥ 0. 

 
Let 𝐻 ⊂ 𝐷𝑜𝑚(𝑥) be the set of neutrosophic hedges, regarded as unary operations. Each 

hedge h∈H is a functor, or comparative particle for adjectives and adverbs as in the natural 
language (English). 

h:Dom(x)→Dom(x)  
x→h(x).      (5) 
Instead of h(x) one easily writes hx to be closer to the natural language. 
By associating the neutrosophic components, one has: 
h_〈t_h,i_h,f_h 〉  x_〈t_x,i_x,f_x 〉 . 
A hedge applied to x may increase, decrease, or approximate the neutrosophic value of the 

element x. 
There also exists a neutrosophic identity I∈Dom(x), denoted I_〈0,0,0〉  that does not hange 

on the elements: 
I_〈0,0,0〉  x_〈t_x,i_x,f_x 〉 . 
In most cases, if a hedge increases / decreases the neutrosophic value of an element x 

situated above the neutral element w, the same hedge does the opposite, decreases / increases the 
neutrosophic value of an element y situated below the neutral element w. 
 

And reciprocally. 
If a hedge approximates the neutrosophic value, by diminishing it, of an element x situated 

above the neutral element w, then it approximates the neutrosophic value, by enlarging it, of an 
element y situated below the neutral element w. 

Let's refer the hedges with respect to the upper part (⊔), above the neutral element, since for 
the lower part (L) it will automatically be the opposite effect. 

We split de set of hedges into three disjoint subsets: 
 H_⊔^+ = the hedges that increase the neutrosophic value of the upper elements; 
 H_⊔^- = the hedges that decrease the neutrosophic value of the upper elements; 
 H_⊔^∼ = the hedges that approximate the neutrosophic value of the upper elements. 
 
Notations: Let 𝑥 = 𝑥⊔ ∪ 𝑤 ∪ 𝑥 , where 𝑥⊔ cons-titutes the upper element set, while 𝑥  the lower 
element subset, 𝑤 the neutral element. 𝑥⊔ and 𝑥  are disjoint two by two. 

3. Operations on Neutrosophic Components 

Let 〈𝑡 , 𝑖 , 𝑓 〉, 〈𝑡 , 𝑖 , 𝑓 〉 neutrosophic numbers. 
Then: 

𝑡 + 𝑡 =
𝑡 + 𝑡 , if 𝑡 + 𝑡 ≤ 1; 

1, if 𝑡 + 𝑡 > 1;
    (6) 

and  



F. Smarandache - Neutrosophic Hedge Algebras 

119 

𝑡 − 𝑡 =
0, if 𝑡 − 𝑡 < 0; 

𝑡 − 𝑡 , if 𝑡 − 𝑡 ≥ 0.
    (7) 

Similarly for 𝑖  and 𝑓 : 

𝑖 + 𝑖 =
𝑖 + 𝑖 , if 𝑖 + 𝑖 ≤ 1; 

1, if 𝑖 + 𝑖 > 1;
    (8) 

𝑖 − 𝑖 =
0, if 𝑖 − 𝑖 < 0; 

𝑖 − 𝑖 , if 𝑖 − 𝑖 ≥ 0.
    (9) 

and  

𝑓 + 𝑓 =
𝑓 + 𝑓 , if 𝑓 + 𝑓 ≤ 1; 

1, if 𝑓 + 𝑓 > 1;
    (10) 

𝑓 − 𝑓 =
0, if 𝑓 − 𝑓 < 0; 

𝑓 − 𝑓 , if 𝑓 − 𝑓 ≥ 0.
    (11) 

 
4. Neutrosophic Hedge-Element Operators 
We define the following operators: 

4.1. Neutrosophic Increment 

Hedge ↑  Element = 〈𝑡 , 𝑖 , 𝑓 〉 ↑ 〈𝑡 , 𝑖 , 𝑓 〉 = 〈𝑡 + 𝑡 , 𝑖 − 𝑖 , 𝑓 − 𝑓 〉,    
  (12) 

meaning that the first triplet increases the second. 

4.2. Neutrosophic Decrement 

Hedge ↓  Element = 〈𝑡 , 𝑖 , 𝑓 〉 ↓ 〈𝑡 , 𝑖 , 𝑓 〉 = 〈𝑡 − 𝑡 , 𝑖 + 𝑖 , 𝑓 + 𝑓 〉,    
  (13) 

meaning that the first triplet decreases the second. 

4.3. Theorem 1 

The neutrosophic increment and decrement operators are non-commutattive. 

5. Neutrosophic Hedge-Hedge Operators 

Hedge ↑  Hedge = 〈𝑡 , 𝑖 , 𝑓 〉 ↑ 〈𝑡 , 𝑖 , 𝑓 〉 = 〈𝑡 + 𝑡 , 𝑖 + 𝑖 , 𝑓 + 𝑓 〉   
  (14) 

Hedge ↓  Hedge = 〈𝑡 , 𝑖 , 𝑓 〉 ↓ 〈𝑡 , 𝑖 , 𝑓 〉 = 〈𝑡 − 𝑡 , 𝑖 − 𝑖 , 𝑓 − 𝑓 〉   
  (15) 

6. Neutrosophic Hedge Operators 

Let 𝑥⊔〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ∈ 𝐷𝑜𝑚(𝑥) i.e. 𝑥⊔ is an upper element of 𝐷𝑜𝑚(𝑥), and 
– ℎ⊔ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ∈ 𝐻⊔ ,  
– ℎ⊔ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ∈ 𝐻⊔ ,  
– ℎ⊔

∽〈𝑡
⊔
∽ , 𝑖

⊔
∽ , 𝑓

⊔
∽ 〉 ∈ 𝐻⊔

∽, 
then ℎ⊔  applied to 𝑥⊔ gives  
(ℎ⊔ 𝑥⊔)〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ↑ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔

〉, 

and ℎ⊔  applied to 𝑥⊔ gives  
(ℎ⊔ 𝑥⊔)〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ↓ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉, 
and ℎ⊔

∽ applied to 𝑥⊔ gives  
(ℎ⊔

∼𝑥⊔)〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉 ↓ 〈𝑡 ⊔∼ , 𝑖 ⊔∼ , 𝑓 ⊔∼ 〉. 



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Now, let 𝑥 〈𝑡 , 𝑖 , 𝑓 〉 ∈ 𝐷𝑜𝑚(𝑥 ), i.e. 𝑥  is a lower element of 𝐷𝑜𝑚(𝑥). Then, ℎ⊔  
applied to 𝑥  gives: 

ℎ⊔ 𝑥 〈𝑡 , 𝑖 , 𝑓 〉 ↓ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔
〉, 

and ℎ⊔  applied to 𝑥  gives: 

ℎ⊔ 𝑥 〈𝑡 , 𝑖 , 𝑓 〉 ↑ 〈𝑡 ⊔ , 𝑖 ⊔ , 𝑓 ⊔ 〉, 
and ℎ⊔

∽ applied to 𝑥  gives: 

ℎ⊔
∽𝑥 〈𝑡 , 𝑖 , 𝑓 〉 ↑ 〈𝑡

⊔
∽ , 𝑖

⊔
∽ , 𝑓

⊔
∽ 〉. 

In the same way, we may apply many increasing, decreasing, approximate or other type of 
hedges to the same upper or lower element  

ℎ⊔ ℎ⊔ ℎ⊔ … ℎ⊔ 𝑥, 
generating new elements in 𝐷𝑜𝑚(𝑥). 
The hedges may be applied to the constants as well. 

6.1. Theorem 2 

A hedge applied to another hedge wekeans or stengthens or approximates it. 

6.2. Theorem 3 

If ℎ⊔ ∈ 𝐻⊔  and 𝑥⊔ ∈ 𝐷𝑜𝑚(𝑥⊔), then ℎ⊔ 𝑥⊔ ≥ 𝑥⊔. 
If ℎ⊔ ∈ 𝐻⊔  and 𝑥⊔ ∈ 𝐷𝑜𝑚(𝑥⊔), then ℎ⊔ 𝑥⊔ ≥ 𝑥⊔. 
If ℎ⊔ ∈ 𝐻⊔  and 𝑥 ∈ 𝐷𝑜𝑚(𝑥 ), then ℎ⊔ 𝑥 ≤ 𝑥 . 
If ℎ⊔ ∈ 𝐻⊔  and 𝑥 ∈ 𝐷𝑜𝑚(𝑥 ), then ℎ⊔ 𝑥 ≥ 𝑥 . 

6.3. Converse Hedges 

Two hedges ℎ  and ℎ ∈ 𝐻 are converse to each other, if ∀𝑥 ∈ 𝐷𝑜𝑚(𝑥), ℎ 𝑥 ≤ 𝑥 is 
equivalent to ℎ 𝑥 ≥ 𝑥. 

6.4. Compatible Hedges 

Two hedges ℎ  and ℎ ∈ 𝐻 are compatible, if ∀𝑥 ∈ 𝐷𝑜𝑚(𝑥), ℎ 𝑥 ≤ 𝑥 is equivalent to 
ℎ 𝑥 ≤ 𝑥. 

6.5. Commutative Hedges 

Two hedges ℎ  and ℎ ∈ 𝐻 are commutative, if ∀𝑥 ∈ 𝐷𝑜𝑚(𝑥), ℎ ℎ 𝑥 = ℎ ℎ 𝑥. Otherwise 
they are called non-commutative. 

6.6. Cumulative Hedges 

If ℎ
⊔
 and ℎ

⊔
∈ 𝐻 , then two neutrosophic edges can be cumulated into one: 

ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 = ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ↑ 〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉. 

  (16) 
Similarly, if ℎ

⊔
 and ℎ

⊔
∈ 𝐻 , then we can cumulate them into one: 

ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 = ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ↑ 〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉. 

 (17) 



F. Smarandache - Neutrosophic Hedge Algebras 

121 

Now, if the two hedges are converse, ℎ
⊔
 and ℎ

⊔
, but the neutrosophic components of the 

first (which is actually a neutrosophic number) are greater than the second, we cumulate them into 
one as follows:      

ℎ
⊔

= ℎ
⊔

ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ↓ 〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉.         (18) 

 
But, if the neutrosophic components of the second are greater, and the hedges are com-

mutative, we cumulate them into one as follows: 

ℎ
⊔

= ℎ
⊔

ℎ
⊔

〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉 ↓ 〈𝑡
⊔

, 𝑖
⊔

, 𝑓
⊔

〉           (19) 

 

7. Neutrosophic Hedge Algebra 

𝑁𝐻𝐴 = (𝑥, 𝐺, 𝐶, 𝐻 ∪ 𝐼, ≤ ) constitutes an abstract algebra, called Neutrosophic Hedge 
Algebra. 
 

7.1. Example of a Neutrosophic Hedge Algebra 𝝉 
Let 𝐺 = {𝑆𝑚𝑎𝑙𝑙, 𝐵𝑖𝑔} the set of generators, repres-ented as neutrosophic generators as 

follows: 
𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉, 𝐵𝑖𝑔〈 . , . , . 〉. 
Let 𝐻 = {𝑉𝑒𝑟𝑦, 𝐿𝑒𝑠𝑠} the set of hedges, repres-ented as neutrosophic hedges as follows: 
𝑉𝑒𝑟𝑦〈 . , . , . 〉, 𝐿𝑒𝑠𝑠〈 . , . , . 〉, 
where 𝑉𝑒𝑟𝑦 ∈ 𝐻⊔  and 𝐿𝑒𝑠𝑠 ∈ 𝐻⊔ . 
𝑥 is a neutrosophic linguistic variable whose domain is 𝐺 at the beginning, but extended by 

generators. 
The neutrosophic constants are 
 𝐶 = 0〈 , , 〉, 𝑀𝑒𝑑𝑖𝑢𝑚〈 . , . , . 〉, 1〈 , , 〉 . 
The neutrosophic identity is 𝐼〈 , , 〉. 
We use the neutrosophic inequality ≤ , and the neutrosophic increment / decrement 

operators previously defined. 
 
Let's apply the neutrosophic hedges in order to generate new neutrosophic elements of the 

neutrosophic linguistic variable 𝑥. 
𝑉𝑒𝑟𝑦 applied to 𝐵𝑖𝑔 [upper element] has a positive effect: 
𝑉𝑒𝑟𝑦〈 . , . , . 〉𝐵𝑖𝑔〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . . , . . , . . 〉 = (𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . , . , . 〉. 
Then: 
𝑉𝑒𝑟𝑦〈 . , . , . 〉(𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . , , . 〉. 
𝑉𝑒𝑟𝑦 applied to 𝑆𝑚𝑎𝑙𝑙 [lower element] has a negative effect: 
𝑉𝑒𝑟𝑦〈 . , . , . 〉𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑆𝑚𝑎𝑙𝑙)〈 . . , . . , . . 〉 =

(𝑉𝑒𝑟𝑦 𝑆𝑚𝑎𝑙𝑙)〈 . , . , . 〉. 
If we compute (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦) first, which is a neutrosophic hedge-hedge operator: 
 
𝑉𝑒𝑟𝑦〈 . , . , . 〉𝑉𝑒𝑟𝑦〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦) 〈 . . , . . , . . 〉 =

(𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦)〈 . , . , . 〉,  
and we apply it to Big, we get: 

(𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦)〈 . , . , . 〉𝐵𝑖𝑔〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . . , . . , . . 〉
= (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦 𝐵𝑖𝑔)〈 . , , . 〉, 

so, we get the same result. 
𝐿𝑒𝑠𝑠 applied to 𝐵𝑖𝑔 has a negative effect: 



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Volume 10, Issue 3 (September, 2019), ISSN 2067-3957 

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𝐿𝑒𝑠𝑠〈 . , . , . 〉𝐵𝑖𝑔〈 . , . , . 〉 = (𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)〈 . . , . . , . 〉 = (𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)〈 . , . , . 〉. 
𝐿𝑒𝑠𝑠 applied to 𝑆𝑚𝑎𝑙𝑙 has a positive effect: 
𝐿𝑒𝑠𝑠〈 . , . , . 〉𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉 = (𝐿𝑒𝑠𝑠 𝑆𝑚𝑎𝑙𝑙)〈 . . , . . , . . 〉 =

(𝐿𝑒𝑠𝑠 𝑆𝑚𝑎𝑙𝑙)〈 . , . , . 〉. 
The set of neutrosophic hedges H is enriched through the generation of new neutrosophic 

hedges by combining a hedge with another one using the neutrosophic hedge-hedge operators. 
Further, the newly generated neutrosophic hedges are applied to the elements of the 

linguistic variable, and more new elements are generated. 
Let's compute more neutrosophic elements: 

𝑉𝐿𝐵 = 𝑉𝑒𝑟𝑦〈 . , . , . 〉𝐿𝑒𝑠𝑠〈 . , . , . 〉𝐵𝑖𝑔〈 . , . , . 〉
= (𝑉𝑒𝑟𝑦 𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)

〈 . , . , . 〉 ↑ 〈 . , . , . 〉 ↓ 〈 . , . , . 〉

= (𝑉𝑒𝑟𝑦 𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)〈 . . , . . , . . 〉 ↓ 〈 . , . , . 〉
= (𝑉𝑒𝑟𝑦 𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)〈 . . , . . , . . 〉 = (𝑉𝑒𝑟𝑦 𝐿𝑒𝑠𝑠 𝐵𝑖𝑔)〈 . , , 〉 

𝑉𝑀 = 𝑉𝑒𝑟𝑦〈 . , . , . 〉𝑀𝑒𝑑𝑖𝑢𝑚〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑀𝑒𝑑𝑖𝑢𝑚)〈 . , . , . 〉 ↑ 〈 . , . , . 〉
= (𝑉𝑒𝑟𝑦 𝑀𝑒𝑑𝑖𝑢𝑚)〈 . , . , . 〉 

𝐿𝑀 = 𝐿𝑒𝑠𝑠〈 . , . , . 〉𝑀𝑒𝑑𝑖𝑢𝑚〈 . , . , . 〉 = (𝐿𝑒𝑠𝑠 𝑀𝑒𝑑𝑖𝑢𝑚)〈 . , . , . 〉 ↓ 〈 . , . , . 〉
= (𝐿𝑒𝑠𝑠 𝑀𝑒𝑑𝑖𝑢𝑚)〈 . , . , . 〉 

𝑉𝑉𝑆 = 𝑉𝑒𝑟𝑦〈 . , . , . 〉𝑉𝑒𝑟𝑦〈 . , . , . 〉𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉 = (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦)〈 . , . , . 〉𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉
= (𝑉𝑒𝑟𝑦 𝑉𝑒𝑟𝑦 𝑆𝑚𝑎𝑙𝑙)〈 . , . , . 〉 

𝑉𝐿𝑆 = 𝑉𝑒𝑟𝑦〈 . , . , . 〉𝐿𝑒𝑠𝑠〈 . , . , . 〉𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉 = 𝑉𝑒𝑟𝑦〈 . , . , . 〉(𝐿𝑒𝑠𝑠 𝑆𝑚𝑎𝑙𝑙)〈 . , . , . 〉
= (𝑉𝑒𝑟𝑦 𝐿𝑒𝑠𝑠 𝑆𝑚𝑎𝑙𝑙)〈 . , . , . 〉 
𝐿𝐴𝑀𝑎𝑥 = 𝐿𝑒𝑠𝑠〈 . , . , . 〉𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑎𝑥𝑖𝑚𝑢𝑚〈 , , 〉

= (𝐿𝑒𝑠𝑠 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑎𝑥𝑖𝑚𝑢𝑚)〈 . , . , . 〉 ↓ 〈 , , 〉
= (𝐿𝑒𝑠𝑠 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑎𝑥𝑖𝑚𝑢𝑚)〈 . , . , . 〉 

𝐿𝐴𝑀𝑖𝑛 = 𝐿𝑒𝑠𝑠〈 . , . , . 〉𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑖𝑛𝑖𝑚𝑢𝑚〈 , , 〉 = (𝐿𝑒𝑠𝑠 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑖𝑛𝑖𝑚𝑢𝑚)〈 . , . , . 〉 ↑ 〈 , , 〉
= (𝐿𝑒𝑠𝑠 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑎𝑥𝑖𝑚𝑢𝑚)〈 . , . , . 〉 

7.2. Theorem 4 
Any increasing hedge ℎ〈 , , 〉 applied to the absolute maximum cannot overpass the absolute 

maximum. 
Proof:  

ℎ〈 , , 〉 ↑ 1〈 , , 〉 = (ℎ1)〈 , , 〉 
= (ℎ1)〈 , , 〉 = 1〈 , , 〉. 

7.3. Theorem 5 
Any decreasing hedge ℎ〈 , , 〉 applied to the absolute minimum cannot pass below the 

absolute minimum. 
Proof:  

ℎ〈 , , 〉 ↓ 0〈 , , 〉 = (ℎ𝑜)〈 , , 〉 
= (ℎ𝑜)〈 , , 〉 = 0〈 , , 〉. 

8. Diagram of the Neutrosophic Hedge Algebra τ 

1〈 , , 〉  ABSOLUTE MAXIMUM 
 
𝑉𝑉𝐵〈 . , , . 〉  Very Very Big 



F. Smarandache - Neutrosophic Hedge Algebras 

123 

𝐿𝐴𝑀〈 . , . , . 〉 Less Absolute Maximum 
𝑉𝐵〈 . , . , . 〉  Very Big 
 
𝐵𝑖𝑔〈 . , . , . 〉  
𝑉𝑀〈 . , . , . 〉 Very Medium 
𝐿𝑉〈 . , . , . 〉  Less Big 
 
𝑉𝐿𝐵〈 . , , 〉  Very Less Big 
𝑉𝐿𝑆〈 . , . , . 〉 Very Less Small 
𝑀〈 . , . , . 〉  MEDIUM 
𝐿𝑀〈 . , . , . 〉 Less Medium 
 
𝐿𝑆〈 . , . , . 〉  Less Small 
𝑆𝑚𝑎𝑙𝑙〈 . , . , . 〉   
𝑉𝑆〈 . , . , . 〉  Very Small 
𝐿𝐴𝑀𝑖𝑛〈 . , . , . 〉 Less Absolute Minimum 
𝑉𝑉𝑆〈 . , . , . 〉 Very Very Small 
 
0〈 , , 〉  ABSOLUTE MINIMUM 
 
9. Conclusions 
In this paper, the classical hedge algebras have been extended for the first time to 

neutrosophic hedge algebras. With respect to an attribute, we have inserted the neutrosophic 
degrees of membership / indeterminacy / nonmembership of each generator, hedge, and constant. 
More than in the classical hedge algebras, we have introduced several numerical hedge operators: 
for hedge applied to element, and for hedge combined with hedge. An extensive example of a 
neutrosophic hedge algebra is given, and important properties related to it are presented. 

 
References  

Cat Ho, N.; Wechler, W. Hedge Algebras: An algebraic Approach to Structure of Sets of Linguistic 
Truth Values. Fuzzy Sets and Systems 1990, 281-293. 

Lakoff, G. Hedges, a study in meaning criteria and the logic of fuzzy concepts. 8th Regional 
Meeting of the Chicago Linguistic Society, 1972. 

Zadeh, L.A. A fuzzy-set theoretic interpretation of linguistic hedges. Journal of Cybernetics 1972, 
Volume 2, 04-34.