Fuzzy quaternion numbers

Abstract

In this paper we build the concept of fuzzy quaternion numbers as a natural extension of fuzzy real numbers. We discuss some important concepts such as their arithmetic properties, distance, supremum, infimum and limit of sequences.

Full text

Fuzzy Quaternion Numbers
Ronildo P. A. Moura , Flaulles B. Bergamaschi ∗, Regivan H. N. Santiago and Benjamin R. C. Bedregal
Department of Informatics and Applied Mathematics
Federal University of Rio Grande do Norte
RN, Brazil, postal code 59072-970
Email: ronildo@ppgsc.ufrn.br, {regivan, bedregal}@dimap.ufrn.br
∗Department of Mathematics and Computer Science
University of southwest of Bahia
BA, Brazil, postal code 45083-900
Email: flaulles@yahoo.com.br
Abstract—In this paper we build the concept of fuzzy
quaternion numbers as a natural extension of fuzzy real
numbers. We discuss some important concepts such as
their arithmetic properties, distance, supremum, infimum
and limit of sequences.
Index Terms—Fuzzy numbers, quaternion, intervals.
I. INTRODUCTION
William Rowan Hamilton proposed first, in 1837, the-
ory of quaternion numbers, in which a complex number
is represented as an ordered pair of real numbers. He
had in mind their interpretation as vectors in the two-
dimensional plane as well the associated algebra, which
would allow him to operate with vectors in the plane.
He was also aware of the greatest problem of his time,
coming from Physics: To construct a language which
would be appropriate to develop the field of Dynamics
in a similar way Newton created Calculus. To achieve
to, it was necessary to create an algebra to manipulate
the vectors.
He noted that it would not be possible to construct
such a structure based on geometrical considerations, but
on operators acting on vectors, more precisely with a
four-dimensional algebra.
He considered elements of the form α=a+bi +cj +
dk, which he called quaternions, where the coefficients
a, b, c, d are real numbers and i, j, k are formal symbols
called basic units. It was obvious to him that two
elements should be added componentwise by formula:
(a+bi +cj +dk) + (a′+b′i+c′j+d′k) = (a+a′) +
(b+b′)i+ (c+c′)j+ (d+d′)k.
The main difficulty was to define the product of two
elements. Since this product should have the usual prop-
erties of a multiplication, such as the distributive law,
it would actually be enough to decide how to multiply
the symbols i, j, k among themselves. This demanded
considerable effort of young Hamilton. He also implicitly
assumed that the product should be commutative. It was
perfectly possible, since he was about to find the first
non-commutative algebra in the entire history of Math-
ematics. Thus, in 1843 he discovered the fundamental
laws of the product of quaternions:
i2=j2=k2=ijk =−1, which also implies the
well-known formulas:
ij =k=−ji,jk =i=−kj,ki =j=−ik.
Afterwards, he presented an extensive memoir on
quaternions to the Royal Irish Academy. His discovery
came as a shock to the mathematicians of the time,
because it opened the possibilities of new extensions of
the field of complex numbers.
In 1989 Buckley [1] gave the first steps toward the ex-
tension of fuzzy real numbers to complex fuzzy numbers.
This paper shows that fuzzy complex numbers is closed
under arithmetic operations and they may be performed
in terms of α-cuts. In 1992 Zhang [2] introduced a
new definition for fuzzy complex numbers. This defini-
tion induced some results analogous to a Mathematical
Analysis. Finally in 2011 Tamir [3] introduced fuzzy
complex numbers with an axiomatic approach. Following
those steps we realized the possibility to extend complex
numbers to quaternions numbers in the same way that
[2] has done. Therefore, this paper propose an extension
for the set of fuzzy real numbers to the set of fuzzy
quaternion numbers. In doing so, we will be able to
understand their features and gather some results which
will allow us to prove results similar to Mathematical
Analysis.
In section 2, we introduce the concept of fuzzy quater-

nion numbers and some initial propositions. Section 3 it
is introduced the interval quaternion numbers. This struc-
ture is important, since the α-cuts of fuzzy quaternion
number will be an interval quaternion number. In section
4, we show that the set of fuzzy quaternion numbers is
partially ordered and we can obtain a metric derived from
the metric of fuzzy real numbers. We also introduce the
concepts of supremum and infimum. Moreover, we take
a closer look at the limit of sequences that is essential to
the study of fuzzy quaternion numbers in the perspective
of Mathematical Analysis.
II. DE FIN IT IO NS A ND BA SI C PROPERTIES
We consider Ras the set of real numbers and Has
the set of quaternion numbers.
Definition 1: A fuzzy real set is a function ¯
A:R−→
[0,1].
Definition 2: A fuzzy real set ¯
Awill be a fuzzy real
number iff:
(1) ¯
Ais normal, i.e., there exists x∈Rwhose ¯
A(x) =
1;
(2) for all α∈(0,1], the set ¯
A[α] = {x∈R:¯
A(x)≥
α}is limited set.
The set of all fuzzy real numbers is denoted RF.
We can see that R⊂RF, since every a∈Rcan be
write as a:R−→ [0,1], where a(x) = 1 if x=aand
a(x) = 0 if x̸=a.
Definition 3: A fuzzy quaternion number is given by
h′:H−→ [0,1] such that h′(a+bi +cj +dk) =
min{¯
A(a),¯
B(b),¯
C(c),¯
D(d)}for some ¯
A, ¯
B, ¯
C, ¯
D∈
RF
The set of all fuzzy quaternion numbers is denoted
by HFand identified as R4
Fwhere every element h′is
associated with (¯
A, ¯
B, ¯
C, ¯
D).
For a better understanding of fuzzy quaternion num-
bers we consider h′= ( ¯
A, ¯
B, ¯
C, ¯
D)∈HFin which
Re(h′) = ¯
Ais called real part and Im1(h′) =
¯
B, I m2(h′) = ¯
C, Im3(h′) = ¯
Dimaginary parts.
Like fuzzy real numbers, H⊂HF,
h=a+bi +cj +dk ∈H,h′:H−→ [0,1]
given by:
h′(x+yi +zj +wk) =
{1if x=aand y=band z=dand w=k,
0if x̸=aor y̸=bor z̸=dor w̸=k,
Proposition 1: For every h′∈HFthere exists s∈H
such that h′(s) = 1,i.e., every fuzzy quaternion number
is normal.
Proof: Let h′∈HF, then h′= ( ¯
A, ¯
B, ¯
C, ¯
D)
¯
A, ¯
B, ¯
C, ¯
D∈RF. Since every fuzzy real number is
normal, then there exist a, b, c, d ∈Rsuch that ¯
A(a) =
¯
B(b) = ¯
C(c) = ¯
D(d)=1. Let s=a+bi +cj +dk so
we conclude that h′(s) = 1.
Let α∈(0,1] and ¯
A∈RF. The set ¯
A[α] = {x∈
R:¯
A(x)≥α}is the α-cut of ¯
A. In the case of fuzzy
quaternion numbers the α-cut will be the set h′[α] =
{s∈H:h′(s)≥α}.
Proposition 2: For every h′∈HFand α∈(0,1],
h′[α] = ¯
A[α]ׯ
B[α]ׯ
C[α]ׯ
D[α],i.e.,h′[α]is a
hiper-cube in R4.
Proof: Let h′= ( ¯
A, ¯
B, ¯
C, ¯
D)and
z=a+bi +cj +dk ∈h′[α], then
h′(z) = min{¯
A(a),¯
B(b),¯
C(c),¯
D(d)} ≥ α
and ¯
A(a),¯
B(b),¯
C(c),¯
D(d)≥α,i.e.,
a∈¯
A[α], b ∈¯
B[α], c ∈¯
C[α]ed∈¯
D[α].
Therefore, z∈¯
A[α]ׯ
B[α]ׯ
C[α]ׯ
D[α].
On the other hand, if z∈¯
A[α]ׯ
B[α]ׯ
C[α]ׯ
D[α],
then z=a+bi+cj +dk,a∈¯
A[α], b ∈¯
B[α], c ∈¯
C[α],
and d∈¯
D[α]. If we continue on the inverse process of
the first part it is easy to conclude that z∈h′[α].
Proposition 3: For every h′∈HF,h′is a
convex fuzzy set, i.e.,h′(λh1+ (1 −λ)h2)≥
min{h′(h1), h′(h2)}where h1, h2∈Hand λ∈[0,1].
Proof: Let h1= (x1, y1, z1, w1),h2=
(x2, y2, z2, w2)and h′= ( ¯
A, ¯
B, ¯
C, ¯
D). We know that
¯
A, ¯
B, ¯
C, ¯
D∈RFare fuzzy real sets. Thus, we have:
¯
A(λx1+ (1 −λ)x2)≥min{¯
A(x1),¯
A(x2)};
¯
B(λy1+ (1 −λ)y2)≥min{¯
B(y1),¯
B(y2)};
¯
C(λz1+ (1 −λ)z2)≥min{¯
C(z1),¯
C(z2)};
¯
D(λw1+ (1 −λ)w2)≥min{¯
D(w1),¯
D(w2)}.
Therefore,
h′(λh1+ (1 −λ)h2) = h′(λ(x1, y1, z1, w1) + (1 −
λ)(x2, y2, z2, w2)) = h′(λx1+ (1 −λ)x2, λy1+
(1 −λ)y2, λz1+ (1 −λ)z2, λw1+ (1 −λ)w2) =
min{¯
A(λx1+ (1 −λ)x2),..., ¯
D(λw1+ (1 −λ)w2)} ≥
min{¯
A(x1),¯
A(x2), . . . , ¯
D(w1),¯
D(w2)} ≥
min{h′(h1), h′(h2)}.
Proposition 4: If h′∈HFand α1, α2∈(0,1] with
α1≤α2, then h′[α1]⊇h′[α2].
Proof: Let z∈h′[α2],z=a+bi +cj +dk
and h′= ( ¯
A, ¯
B, ¯
C, ¯
D), according to proposition 2
a∈¯
A[α2], b ∈¯
B[α2], c ∈¯
C[α2]and d∈¯
D[α2]. As
we can see that α1≤α2, then ¯
A[α1]⊇¯
A[α2],¯
B[α1]⊇
¯
B[α2],¯
C[α1]⊇¯
C[α2]and ¯
D[α1]⊇¯
D[α2]. Thus,
a∈¯
A[α1], b ∈¯
B[α1], c ∈¯
C[α1]ed∈¯
D[α1]and
z∈h′[α1].

In HFwe can define addition and multiplication as in
[4].
Definition 4: Let s′, h′∈HFwhere
s′= ( ¯
X, ¯
Y , ¯
Z, ¯
W)and h′= ( ¯
A, ¯
B, ¯
C, ¯
D), then:
s′+h′= ( ¯
X+¯
A, ¯
Y+¯
B, ¯
Z+¯
C, ¯
W+¯
D),
s′·h′= ( ¯
X¯
A−¯
Y¯
B−¯
Z¯
C−¯
W¯
D, ¯
X¯
B+¯
Y¯
A+¯
Z¯
D−
¯
W¯
C, ¯
X¯
C−¯
Y¯
D+¯
Z¯
A+¯
W¯
B, ¯
X¯
D−¯
Y¯
C−¯
Z¯
B+¯
W¯
A).
Proposition 5: For all s′, h′, t′∈HF:
(1) s′+h′∈HF;
(2) s′+h′=h′+s′;
(3) (s′+h′) + t′=s′+ (h′+t′);
(4) there is a 0′= (¯
0,¯
0,¯
0,¯
0) where h′+ 0′=h′for
all h′∈HF.
Proof: According to [4], these properties are valid
for fuzzy real numbers. Thus, it is also valid for fuzzy
quaternion numbers.
Proposition 6: For all s′, h′, t′∈HF:
(1) s′·h′∈HF;
(2) (s′·h′)·t′=s′·(h′·t′);
(3) there is a 1′= (¯
1,¯
0,¯
0,¯
0) where h′·1′=h′for all
h′∈HF.
Proof: Immediately.
III. INT ERVAL QUATERNION NUMBER
In this section we introduce the concept of interval
quaternion number and its properties, since the α-cuts
of fuzzy quaternion numbers will be interval quaternion
numbers. We also aim to prove some results that will be
useful in the next section such as the density of HF.
Consider the set of closed intervals I(R) =
{[a, b] : a, b ∈R}endowed with the following
arithmetic:[5]
1) [a, b]+[c, d] = [a+c, b +d];
2) [a, b]−[c, d] = [a−d, b −c], in wich
−[c, d] = [−d, −c];
3) [a, b]·[c, d] = [min{a·c, a ·d, b ·c, b ·d}, max{a·
c, a ·d, b ·c, b ·d}];
4) [c, d]−1=1
[c, d]= [1
d,1
c], if 0̸∈ [c, d];
5) [a, b]
[c, d]= [a, b]·[c, d]−1, whenever 0̸∈ [c, d].
Definition 5: An interval quaternion number His a
tuple (A, B, C, D), where A, B, C, D ∈I(R). The set
of all interval quaternion numbers is I(H).
Definition 6: Let S= (X, Y, Z, W ),H=
(A, B, C, D)∈I(H), then S=Hiff (X=A)∧(Y=
B)∧(Z=C)∧(W=D).
Definition 7: An I(H)we define addition and
multiplication likewise the interval real numbers [5].
Let S, H ∈I(H)where S= (X, Y, Z, W )and
H= (A, B, C, D), then:
S+H= (X+A, Y +B, Z +C, W +D),
S·H= (XA −Y B −ZC −W D, X B +Y A +ZD −
W C, X C −Y D +Z A +W B, X D −Y C −ZB +W A).
Proposition 7: For all S, H, T ∈I(H):
(1) S+H∈I(H);
(2) S+H=H+S;
(3) (S+H) + T=S+ (H+T);
(4) there is a 0∗= (0,0,0,0) where H+ 0∗=Hand
0 = [0,0] ∈I(R).
Proof: Immediately.
Proposition 8: For all S, H ∈I(H):
(1) S·H∈I(H);
(2) there is a 1∗= (1,0,0,0) where H·1∗=Hand
1 = [1,1] ∈I(R).
Proof: Immediately.
We refer to 0and 1in I(H)instead of 0∗and 1∗.
Observe that we neither have the inverse of addition nor
the inverse of multiplication.
Observation 1: We have no commutativity and as-
sociativity of multiplication in the interval quaternion
numbers I(H). This statement is not hard to prove.
Definition 8: (metric) Let S, H ∈I(H)where S=
(X, Y, Z, W )and H= (A, B, C, D). If dis a metric in
I(R), then we define D(S, H) = d(X, A) + d(Y, B ) +
d(Z, C) + d(W, D).
It’s not hard to prove that Dis a metric in I(H), since
dis a metric in I(R).
Definition 9: Let S, H ∈I(H)where S=
(X, Y, Z, W )and H= (A, B, C, D), then S≤H⇔
(X≤A)∧(Y≤B)∧(Z≤C)∧(W≤D)and
S < H ⇔(X < A)∧(Y < B)∧(Z < C )∧(W < D).
Clearly, ≤is a partial order in I(H).
Definition 10: A non-empty set Cis said to be dense,
if for every a, b ∈Cand a < b there exists a c∈Cwith
a<c<b.
Proposition 9: I(H)is dense.
Proof: Let S < H, then (X < A)∧(Y < B)∧(Z <
C)∧(W < D). We know that I(R)is dense. Thus, there
exist T1, T2, T3, T4∈I(R)where (X < T1< A)∧(Y <
T2< B)∧(Z < T3< C )∧(W < T4< D). Therefore,
S < T < H where T= (T1, T2, T3, T4).

We close this section with two propositions about the
α-cuts of fuzzy quaternion numbers which establish a
strong connection between I(H)and HF.
Proposition 10: For every h′∈HFand α∈(0,1],
then h′[α]⊂I(H).
Proof: Immediate from previous definitions and
proposition 2.
Proposition 11: Let s′, h′∈HF, where s′=
(¯
X, ¯
Y , ¯
Z, ¯
W),h′= ( ¯
A, ¯
B, ¯
C, ¯
D)and α∈(0,1]. Hence:
(1) (s′+h′)[α] = s′[α] + h′[α];
(2) (s′·h′)[α] = s′[α]·h′[α].
Proof:
(1) Considering h′[α] = ( ¯
A[α],¯
B[α],¯
C[α],¯
D[α]) we
have:
(s′+h′)[α] = (( ¯
X+¯
A)[α],(¯
Y+¯
B)[α],(¯
Z+
¯
C)[α],(¯
W+¯
D)[α]) = ( ¯
X[α]+ ¯
A[α],¯
Y[α]+ ¯
B[α],¯
Z[α]+
¯
C[α],¯
W[α] + ¯
D[α]) = ( ¯
X[α],¯
Y[α],¯
Z[α],¯
W[α]) +
(¯
A[α],¯
B[α],¯
C[α],¯
D[α]) = s′[α] + h′[α],
(2) Analogous.
IV. METRIC FOR FU ZZ Y QUATERNION NUMBERS AND
TH EI R PRO PE RTI ES
In this section we will have a look at HFas a partially
ordered set. For the next definition, we will consider ≤I
as any partial order in I(R).
Definition 11: Let ¯
A, ¯
B∈RF. We say that ¯
A≤¯
B
iff ¯
A[α]≤I¯
B[α]for all α∈(0,1].¯
A < ¯
Biff ¯
A≤¯
B
and there exists α0∈(0,1] such that ¯
A[α0]<¯
B[α0].
Note that ¯
A=¯
Biff ¯
A≤¯
Band ¯
B≤¯
A.
Definition 12: We say that ¯
Ais a infinite fuzzy real
number iff for all M∈R+there exists αM∈(0,1] such
that [−M, M ]⊆A[αM]. We denote ¯
A= ¯∞.
Definition 13: Let h′= ( ¯
A, ¯
B, ¯
C, ¯
D)∈HF. We say
that h′is a infinite fuzzy quaternion number iff ¯
A= ¯∞
or ¯
B= ¯∞or ¯
C= ¯∞or ¯
D= ¯∞. We denote h′=∞′.
Definition 14: Let s′= ( ¯
X, ¯
Y , ¯
Z, ¯
W)and h′=
(¯
A, ¯
B, ¯
C, ¯
D). We say that s′≤h′⇔(( ¯
X≤¯
A)∧(¯
Y≤
¯
B)∧(¯
Z≤¯
C)∧(¯
W≤¯
D)) and s′< h′⇔(( ¯
X <
¯
A)∧(¯
Y < ¯
B)∧(¯
Z < ¯
C)∧(¯
W < ¯
D)).
We can easily see ≤as a partial order. The next
proposition tells us an important fact according definition
to 10.
Proposition 12: HFis dense.
Proof: Let s′< h′and s′= ( ¯
X, ¯
Y , ¯
Z, ¯
W)and
h′= ( ¯
A, ¯
B, ¯
C, ¯
D). Then (( ¯
X < ¯
A)∧(¯
Y < ¯
B)∧(¯
Z <
¯
C)∧(¯
W < ¯
D)). Thus, ¯
X[α]<¯
A[α]for all α∈(0,1].
We know that I(H)is dense, hence there exists ¯
T1[α]
with ¯
X[α]<¯
T1[α]<¯
A[α]for each α∈(0,1]. To
make the same happen to ¯
B, ¯
C, ¯
Dwe obtain the fuzzy
quaternion number f′= ( ¯
T1,¯
T2,¯
T3,¯
T4)s′< f′< h′.
Proposition 13: Given a function d=RF×RF−→
R+, a function D=HF×HF−→ R+, defined as:
D(s′, h′) = d(¯
X, ¯
A) + d(¯
Y , ¯
B) + d(¯
Z, ¯
C) + d(¯
W , ¯
D),
where s′= ( ¯
X, ¯
Y , ¯
Z, ¯
W)and h′= ( ¯
A, ¯
B, ¯
C, ¯
D)is a
metric, whenever dis a metric.
Proof:
If d(x, y)≥0, then D(s′, h′)≥0.
D(s′, h′) = d(¯
X, ¯
A) + d(¯
Y , ¯
B) + d(¯
Z, ¯
C) + d(¯
W , ¯
D)
=d(¯
A, ¯
X) + d(¯
B, ¯
Y) + d(¯
C, ¯
Z) + d(¯
D, ¯
W)
=D(h′, s′)
Clearly, D(s′, h′)=0⇔d(¯
X, ¯
A) = d(¯
Y , ¯
B) =
d(¯
Z, ¯
C) = d(¯
W , ¯
D)=0⇔¯
X=¯
A, ¯
Y=¯
B, ¯
Z=
¯
C, ¯
W=¯
D⇔s′=h′.
D(s′, h′) = d(¯
X, ¯
A) + d(¯
Y , ¯
B) + d(¯
Z, ¯
C) + d(¯
W , ¯
D)≤
d(¯
X, ¯
F1)+d(¯
F1,¯
A)+d(¯
Y , ¯
F2)+d(¯
F2,¯
B)+d(¯
Z, ¯
F3)+
d(¯
F3,¯
C)+ d(¯
W , ¯
F4)+ d(¯
F4,¯
D)≤D(s′, f ′)+ D(f′, h′)
where f′= ( ¯
F1,¯
F2,¯
F3,¯
F4).
Corollary 1: ⟨HF, D⟩is a metric space.
Proof: Immediately.
Definition 15: Let R⊆HF. If there exist M′∈
HF, M ′̸=∞′, such that c′≤M′, for every c′∈R,
then Ris said to have an upper bound M′. Similarly, if
there exist m′∈HF, m′̸=∞′, such that m′≤c′, for
every c′∈R, then Ris said to have a lower bound.
A set with both upper and lower bounds is said to be
bounded.
Definition 16: We say that s′∈HFis the least upper
bound for R⊆HFif s′has the following properties:
(1) c′≤s′for all c′∈R;
(2) for any ϵ > 0, ϵ ∈Rthere exists c′∈Rsuch that
s′< c′+ϵ.
We write s′=sup R.
Definition 17: We say that i′∈HFis the greatest
lower bound for R⊆HFif i′has the following
properties:
(1) i′≤c′for all c′∈R;
(2) for any ϵ > 0, ϵ ∈Rthere exists c′∈Rsuch that
c′−ϵ < i′.
We write i′=inf R.
Let R⊆HF, we can define ReR={Re(c′)∈
RF:c′∈R},Im1R={Im1(c′)∈RF:c′∈R},
Im2R={Im2(c′)∈RF:c′∈R},I m3R=
{Im3(c′)∈RF:c′∈R}. Thus, R=ReR×Im1R×
Im2R×Im3R.

Proposition 14: If R⊆HFhas the least upper bound,
then:
sup(R) =
(sup(ReR), sup(Im1R), sup(Im2R), sup(I m3R)).
Proof: First of all, we observe that R=ReR×
Im1R×Im2R×I m3Rand, if Rhas the least up-
per bound, then there exist the least upper bounds for
ReR, Im1R, Im2R, I m3R.
(1) If c′∈R, then Re(c′)≤sup(ReR),
Im1(c′)≤sup(Im1R),I m2(c′)≤sup(Im2R),
Im3(c′)≤sup(Im3R)and thus c′≤
(sup(ReR), sup(Im1R), sup(Im2R), sup(I m3R)).
(2) Let ϵ > 0, ϵ ∈Rso there exist ¯
A∈ReR,
¯
B∈Im1R,¯
C∈Im2R,¯
D∈Im3Rwhere sup(ReR)<
¯
A+ϵ,sup(Im1R)<¯
B+ϵ,sup(Im2R)<¯
C+ϵ,
sup(Im3R)<¯
D+ϵ. Have c′= ( ¯
A, ¯
B, ¯
C, ¯
D)and thus
(sup(ReR), sup(Im1R), sup(Im2R), sup(I m3R)) <
c′+ϵ.
Proposition 15: If R⊆HFhas the greatest lower
bound, then:
inf(R) =
(inf(ReR), inf (Im1R), inf (Im2R), inf (Im3R)).
Proof: Similar to the latter proposition.
V. L IMIT OF A SEQUENCE OF FUZZY QUATE RN ION
NUMBERS
In this section we introduce the concept of limit in
HFas in [2]. The next results are straightforward but
relevant if we want to derive deep results in the future.
Definition 18: Given a metric don RF,{¯
An} ⊂ RF
and ¯
A∈RF, the sequence {¯
An}is said to converge to ¯
A
with respect to d, i.e. lim
n→∞
¯
An=¯
A, if for arbitrary ϵ >
0, there exists an integer N > 0such that d(¯
An,¯
A)< ϵ
as n≥N.
Definition 19: Given a metric don RF,{h′
n} ⊂ HF,
¯
h′∈HF. Then {h′
n}is said to converge to h′,lim
n→∞ h′
n=
h′, if for arbitrary ϵ > 0, there exists an integer N > 0
such that
D(h′
n, h′)< ϵ as n≥N.
Theorem 2: lim
n→∞ h′
n=h′iff lim
n→∞ Re(h′
n) = Re(h′)
and lim
n→∞ Im1(h′
n) = Im1(h′)and lim
n→∞ Im2(h′
n) =
Im2(h′)and lim
n→∞ Im3(h′
n) = Im3(h′).
Proof: Immediately.
Theorem 3: Let {¯
An},{¯
Bn} ⊂ RF,¯
A, ¯
B∈RF, a ∈
R. If lim
n→∞
¯
An=¯
Aand lim
n→∞
¯
Bn=¯
B, then:
(1) lim
n→∞
¯
An+¯
Bn=¯
A+¯
B;
(2) lim
n→∞
¯
An−¯
Bn=¯
A−¯
B;
(3) lim
n→∞ a·¯
An=a·¯
A.
Theorem 4: Let {h′
n},{s′
n} ⊂ HFand h′, s′∈HF.
If lim
n→∞ h′
n=h′and lim
n→∞ s′
n=s′, then:
(1) lim
n→∞(h′
n±s′
n) = h′±s′;
(2) lim
n→∞(c·h′
n) = c·h′.
Proof: (1) Since h′
n= ( ¯
An,¯
Bn,¯
Cn,¯
Dn)and s′
n=
(¯
Xn,¯
Yn,¯
Zn,¯
Wn), where ¯
An=Re(h′
n)etc. Thus,
lim
n→∞(h′
n±s′
n) = lim
n→∞(¯
An±¯
Xn,· · · ,¯
Dn±¯
Wn)
= ( ¯
A±¯
B, ··· ,¯
D±¯
W)
=h′±s′
;
(2) Analogous.
Theorem 5: (Limit uniqueness theorem) If lim
n→∞ h′
n=
h′and lim
n→∞ h′
n=s′, then h′=s′.
Proof: Immediately.
Theorem 6: (Sandwich theorem) Let
{h′
n},{t′
n},{s′
n} ⊂ HF,h′∈HF. If for every n,
h′
n≤t′
n≤s′
nand lim
n→∞ h′
n= lim
n→∞ s′
n=h′, then
lim
n→∞ t′
n=h′.
Proof: Immediately.
Theorem 7: (Boundedness theorem). Let {h′
n} ⊂ HF,
h′̸=∞′, h′̸=∞′. If {h′
n}converge, then there exist
L′, l′(̸=∞′)such that l′≤h′
n≤L′for every n.
Proof: Immediately
Theorem 8: Let lim
n→∞ h′
n=h′and lim
n→∞ s′
n=s′.
Then, lim
n→∞ D(h′
n, s′
n) = D(h′, s′).
Proof: Immediately
VI. FI NAL REMARKS
We built a structure which comprises the fuzzy quater-
nion numbers similar to [2], once we understand that
this could be the most natural way to extend the fuzzy
complex numbers. We were able to provide some essen-
tial elements on the development of fuzzy Mathematical
Analysis in the case of fuzzy quaternion numbers. Also,
we checked the existence of partial orders, metrics,
the concept of supremum, and limit of sequences. In
addition, we established a natural connection from the
set of α-cuts to set of interval quaternion numbers. This
latter fact gives us the computational support for future
studies. Also we can think an axiomatic approach for
fuzzy quaternions numbers as future work.
ACK NOWLEDGME NT S
The authors would like to thank UESB (University of
southwest of Bahia), UFRN (Federal University of Rio
Grande do Norte) and CAPES (Brazilian Agency) for
their financial support.

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