Computational Intelligence - August 2015 - 44

In the following we summarize a hierarchy of complete lattices in seven steps and define certain inclusion measure functions.
Step-1. We assume a totally-ordered, complete lattice ^L, #h
of real numbers, where L 3 R = R , {- 3, + 3} with least
and greatest elements denoted by o and i, respectively. The corresponding inf ^/ h and sup ^0 h operators are the min and the
max operators, respectively. In lattice ^L, #h we consider both a
strictly increasing function v : L " [0, 3 h, such that v (o) = 0 as
well as v ^ i h < + 3, and a strictly decreasing function i : L " L,
such that i ^ o h = i as well as i ^ i h = o.
Step-2. We assume the partially-ordered, complete lattice
^I 1, 3 h of (T1) intervals in lattice ^L, # h . The corresponding
inf ^+ h and sup ^,o h operations are given by [a, b] + [c, d ]
.
= 6a 0 c, b / d@ and [a, b ] , [c, d ] = [a / c, b 0 d ], respectively.
We remark that if a 0 c % b / d then, by definition, [a, b] + [c, d ]
equals the empty set (4) . Two inclusion measure functions
.
v + : I 1 # I 1 " [0, 1] and v , : I 1 # I 1 " [0, 1] are given in lattice
^I 1, 3 h based on a length function V : I 1 " [0, 3) as follows.
1,
for x = 4.
v + (x, y) = * V (x + y)
(1)
, for x 2 4.
V (x )
Z
.
for x , y = 4.
]] 1,
.
.
v , (x, y) = [ V ( y)
(2)
] V (x ,. y) , for x , y 2 4.
\
Recall that a length function V : I 1 " [0, 3) is defined as
V ^x = [a 1, a 2] h = '

0,
x =4
,
v ^i ^a 1hh + v ^a 2h, x 2 4

where functions v (.) and i (.) are as in Step-1.
Step-3. We assume the partially-ordered, complete lattice
^I 2, 3 h of T2 intervals in lattice ^I 1, 3 h - Recall that a T2 interval
is defined as an interval of T1 intervals. The corresponding inf ^+ h
and sup ^,o h operations are given by [[a 1, a 2 ], [b 1, b 2 ]] + [[c 1, c 2],
[d 1, d 2 ]] = [[a 1 / c 1, a 2 0 c 2], [b 1 0 d 1, b 2 / d 2 ]], and [[a 1, a 2], [b 1,
b 2]] ,o [[c 1, c 2],[d 1, d 2]] = [[a 1 0 c 1, a 2 / c 2],[b 1 / d 1, b 2 0 d 2]], respectively. We remark that if [a 1 / c 1, a 2 0 c 2] M [b 1 0 d 1, b 2 / d 2] then,
by definition, [[a 1, a 2], [b 1, b 2]] + [[c 1, c 2], [d 1, d 2]] equals the empty
set (4) . Two inclusion measure functions v + : I 2 # I 2 " [0, 1] and
.
v , : I 2 # I 2 " [0, 1] are given in lattice ^ I 2, 3 h based on a length
function V : I 2 " [0, 3) as follows.
3 [[c 1, c 2], [d 1, d 2]]h
Z 1,
b1 > b2 .
]
b 1 # b 2, b 1 0 d 1 > b 2 / d 2 .
] 0,
(3)
b 1 # b 2, b 1 0 d 1 # b 2 / d 2,
= ] 0,
[
[
,
]
[
a
c
a
c
b
d
/
0
0
,
b
/
d
]
.
M
2
2
1
1
2
2
1
1
]
] V ^[[a 1, a 2], [b 1, b 2]] + [[c 1, c 2], [d 1, d 2]]h
, otherwise.
]
V ^[[a 1, a 2], [b 1, b 2]]h
\

v + ^ [[a 1, a 2], [b 1, b 2]]

v , ^ [[a 1, a 2], [b 1, b 2]]

3 [[c 1, c 2], [d 1, d 2]]h
Z 1,
b1 > b2 .
]
b 1 # b 2, d 1 > d 2 .
= ] 0,
[
V ^[[c 1, c 2], [d 1, d 2]]h
, otherwise.
]
] V ([[a 1, a 2], [b 1, b 2]] ,. [[c 1, c 2], [d 1, d 2]])
\

.

44

IEEE ComputatIonal IntEllIgEnCE magazInE | august 2015

(4)

Recall that a length function V : I 2 " [0, 3) is defined as
V ^x = [[a 1, a 2], [b 1, b 2]] h
0,
x = 4.
='
v ^a 1h + v ^i ^a 2hh + v ^i ^b 1hh + v ^b 2h, x 2 4.

where functions v (.) and i (.) are as in Step-1.
Step-4. We assume the partially-ordered lattice ^F1, ) h of
(T1) Intervals' Numbers, or (T1) INs for short. Recall that
an IN is defined as a function F : [0, 1] " I 1 that satisfies
both h 1 # h 2 & Fh1 4 Fh2 and 6X 3 [0, 1] : + h ! X Fh = F * X .
In particular, an "interval (T1) IN F_ is defined such that
Fh = [a, b], 6h ! [0, 1]; in other words, the aforementioned
interval (T1) IN F ! F1 represents the interval [a, b ] ! I 1 .
An IN is interpreted as an information granule [12]. An IN
F can equivalently be represented either by a set of intervals
Fh, h ! [0, 1] (an IN's interval-representation), or by a function
F (x) = * {h : x ! Fh} (an IN's membership-functionh ! [0, 1]

representation). For F, G ! F1 we have
F ) G + ^6h ! [0, 1] : Fh 3 G hh + ^6x ! L : F (x) # G (x)h.
(5)
The height hgt (F ) of an IN F is defined as the supremum
of its membership function values, i.e., hgt (F ) = * F (x) . The

corresponding inf (O) and sup ^P h operations in lattice
^F1, ) h are given by (F O G ) h = Fh + G h and (F P G )h =
.
Fh , G h, respectively, for h ! [0, 1] . Next, we define two inclusion measure functions v O : F1 # F1 " [0, 1] and v P : F1 #
F1 " [0, 1] based on the inclusion measure functions v + : I 1 #
I 1 " [0, 1] and v ,. : I 1 # I 1 " [0, 1], respectively.
x!L

1

v O (E, F ) =

#

v + (E h, Fh) dh.

(6)

v , (E h, Fh) dh.

(7)

0
1

v P (E, F ) =

#

.

0

Specific advantages of an inclusion measure function in a
Fuzzy Inference System (FIS) context have been reported [13].
Step-5. We assume the partially-ordered, complete lattice
^F2, ) h of T2 INs - Recall that a T2 IN is defined as an
interval of T1 INs; that is, a T2 IN by definition equals
[U, W ] 0 {X ! F1: U ) X ) W }, where U is called lower
IN, and W is called upper IN (of the T2 IN [U, W ]). In the
latter sense we say that X is encoded in 6U, W @ . The corresponding inf (O) and sup ^P h operations in lattice ^F2, ) h are
.
given by ^F O G hh = Fh + G h and ^F P G hh = Fh , G h, respectively. We can define two inclusion measure functions
v O : F2 # F2 " [0, 1] and v P : F2 # F2 " [0, 1], based on the
inclusion measure functions v + : I 2 # I 2 " [0, 1] and
.
v , : I 2 # I 2 " [0, 1], using equations (6) and (7), respectively.
The computation of the join and meet operations in the lattice
^F2, ) h is demonstrated next.
Consider the two T2 INs [ f, F ] and [ g, G ] shown in Fig. 1(a),
where f, F, g, G ! F1 such that f ) F and g ) G. The (join) T2
IN [ f, F ] P [ g, G ] = [ f O g, F P G] is shown in Fig. 1(b), where



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