Algorithm 5 – Lattice
2.4 Continuity and semicontinuity
In our development of a fuzzy extension of FCA we shall need some ideas concerning continuity and semicontinuity of t-norms and implications. In this section we shall define continuity, semicontinuity and find how semicontinuity of a t-norm relates to its residuum. Our definitions of semicontinuity are in accordance with (BOURBAKI, 1966).
Definition 2.4.1. Let p P N. The distance between two points x ✏ ♣x1, ..., xpq, y ✏
♣y1, ..., ypq PRp is defined as
dp♣x, yq ✏
❞ p
➳
i✏1
♣xi✁yiq2 . In particular, d0♣x, yq ✏ 0.
Definition 2.4.2. Given a point xPRp and r→0, theopen ball of centre x and radius r is the set Br♣xq :✏ ty P Rp : dp♣x, yq ➔ r✉. A set S ❸Rp is open iff it is the (arbitrary) union of open balls. A neighbourhood of a set A❸ Rp is a set V such that A ❸S ❸ V, such that S is an open set. In particular,S is a neighbourhood ofA. A neighbourhood of a point a is a neighbourhood of A✏ ta✉.
In other words, an open ball of centre xis the collection of those points close to x (according to the distance dp), and a neighbourhood of a set (or a point) A corresponds to the "surroundings" of A.
Definition 2.4.3. A sequence on a set X is a map N ÑX. If for each n P N we have n ÞÑxn we write♣xnqor ♣xnqnPN.
Now, suppose that X is an ordered set. If for all nP Nwe have xn↕xn 1 we say that the sequence is increasing. If it additionally happens for all n PN thatxn✘xn 1
— i.e., xn➔xn 1 — the we say that the sequence is strictly increasing.
Decreasing and strictly decreasing sequences are defined dually.
Definition 2.4.4. We say that a sequence ♣xnq on Rp converges to x if for every open ball B containing x, there exists n0 P N such that xn P B for all n ➙n0, and we write xn Ñx or lim
nÑ✽xn ✏x.
If p✏1 so thatRp is (linearly) ordered and♣xnqis decreasing (resp. increasing) such that xnÑx, we may write xn ×x (resp. xn Õx).
Example 2.4.5. Consider the following sequences on R.
1. The sequence (see Fig. 10a) ✂ 1 n 1
✡
is a strictly decreasing sequence that converges to 0, i.e., 1④♣n 1q×0.
2. The sequence (see Fig. 10b)
♣ln♣n 1qq
is an increasing diverging14 sequence, where ln is the natural logarithm.
3. The sequence (see Fig. 10c) ✄ n
➳
k✏0
♣✁1qk④k 1
☛
nPN
is neither increasing nor decreasing, yet it converges to ln 2.
Definition 2.4.6. Let X ❸Rp andY ❸Rq be non-empty (classical) sets. Letf :X ÑY be a map. We say that f is continuous at xPX if, for all ε→0, there exists a δ→0 such that for all yPX,
dp♣x, yq ➔ δ implies dq♣f♣xq, f♣yqq ➔ ε .
In other words, f is continuous at x if, for y P X sufficiently close to x — distance less than δ —, the points f♣xq, f♣yqare close in Y — less than ε apart. If f is continuous at every xPX it is said to be continuous.
In our investigation based on t-norms, t-conorms and implications, we have X ✏ r0,1s2 and Y ✏ r0,1s. We shall denoted2 byd and d1♣x, yq by⑤x✁y⑤.
Example 2.4.7. The maximum t-conorm ▽
M is continuous. Consequently, the minimum t-norm △
M is also continuous. In fact, let x✏ ♣x1, x2q P r0,1s2, ε→0 and take δ ✏ε. Let
14 It is unbounded above, i.e., for all k P N there is an n P N such that ln♣n 1q → k. A sequence
♣xnqis calledunbounded below iff♣✁xnqis unbounded above. A sequence isunbounded iff it is either unbounded above or below (or both).
0 5 10 15 20 25 30 0.0
0.2 0.4 0.6 0.8 1.0
(a) Strictly Decreasing
0 5 10 15 20 25 30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
(b) Logarithm
0 5 10 15 20 25 30
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
(c) Partial Sum
Figure 10 – Sequences
y ✏ ♣y1, y2q P r0,1s2 be such that d♣x, yq ➔δ. Definemx :✏x1▽
Mx2 and my :✏y1▽
My2. Since mx, my ➙0 we have
x1y1 ↕mxy1 ↕mxmy .
Similarly, x2y2 ↕mxmy, and so adding and multiplying by -2 we have
✁4mxmy ↕ ✁♣2x1y1 2x2y2q . Thus,
♣⑤mx✁my⑤q2 ✏ ♣mx✁myq2
↕2♣mx✁myq2
✏2m2x 2m2y✁4mxmy
↕ ♣x21 x22q ♣y12 y22q ✁ ♣2x1y1 2x2y2q
✏ ♣x21 y12✁2x1y1q ♣x22 y22✁2x2y2q
✏ ♣x1✁y1q2 ♣x2✁y2q2
✏d♣x, yq2
➔δ2 ✏ε2 .
Since ε →0, taking the square root on both sides we get
⑤mx✁my⑤ ➔ε , and so ▽
M is continuous at x. But x is arbitrary, hence ▽
M is continuous.
Now, according to (2.3.7), x▽
My✏1✁ r♣1✁xq△
M♣1✁yqs , that is, △
M is a composition of continuous functions (the continuous ▽
M and differences), hence it is continuous.
Continuity of a t-norm does not imply continuity of its residuum — for example, the residuum of the continuous minimum t-norm is the discontinuous Gödel implication15. Nonetheless, we can say something about such implications. The following definition is made by (BOURBAKI, 1966).
Definition 2.4.8. Let X ❸ Rn, Y ❸ R and f : X Ñ Y. We say that f is lower semicontinuous at x0 PX if for h such that h➔f♣x0q there is a neighbourhood V of x0
such that
h➔f♣xqfor all xP V .
If f is upper semicontinuous at each x0 P X we say that it isupper semicontinuous.
The notion of upper semicontinuity (either at a point, or of the whole function) is defined dually16.
Notice that f is lower semicontinuous iff ✁f is upper semicontinuous17. Fur-thermore, see that a map is continuous at x0 iff it is both lower and upper semicontinuous at x0.
BOURBAKI also proves18 the following.
Proposition 2.4.9. A function f :X ❸Rn ÑR is lower semicontinuous at x0 PX iff lim inf
nÑ✽ f♣xnq ➙ f♣x0q
15 Indeed, ñg is discontinuous at ♣x, xq for all x P r0,1r. In fact, let ε ✏ 1✁x. Then for all δ → 0,
♣x, x✁δ④2q PBδ♣x, xq, but
✞✞✞✞♣xñgxq ✁
✂ xñg
✂ x✁δ
2
✡✡✞✞✞✞ ✏✞✞
✞✞1✁
✂ x✁δ
2
✡✞✞✞✞ ✏✒✂
1 δ
2
✡
✁x
✚
→1✁x✏ε .
16 I.e., replace each instance of➔by→.
17 Where✁f is the mapxÞÑ ✁f♣xq.
18 In fact, Bourbaki’s result is more general, but Prop. 2.4.9follows when considering theFréchet filter onN, i.e., the complements of finite subsets ofN.
for each sequence ♣xnqnPN converging to x0, where lim inf
nÑ✽ kn :✏ inf
n→0
✂ sup
m➙n
km
✡
for any sequence ♣knqnPN.
Because of the duality between lower and upper semicontinuousness19, we have the following.
Corollary 2.4.10. A function f :X ❸RnÑR is upper semicontinuous at x0 PX iff lim sup
nÑ✽ f♣xnq ↕f♣x0q for each sequence ♣xnqnPN converging to x0, where
lim sup
nÑ✽ kn:✏sup
n→0
✁inf
m➙nkm
✠
for any sequence ♣knqnPN.
We now derive a useful expression concerning lower semicontinuous t-norms in Lemma 2.4.14, and for its proof we use Proposition 2.4.11.
Proposition 2.4.11. Let △ be a lower semicontinuous t-norm and x, y P r0,1s. Let A✏ tzP r0,1s:x△z ↕y✉ .
Then supAPA.
Proof. Let z0 ✏supA and ♣znq be a strictly increasing sequence on r0,1s that converges to z0. By definition of z0 we have
x△zm ↕y for all m→0 , as zm ➔z0 for each m →0. Now, for each n→0 we have
minf➙n♣x△zmq ↕x△zn↕y , so that
sup
n→0
✁inf
m➙n♣x△znq✠
↕y . By lower semicontinuity of △,
x△z0 ↕lim infnÑ✽ ♣x△znq ✏sup
n→0
✁inf
m➙n♣x△zmq✠
↕y . Therefore,z0 P A.
19 In the sense that f is lower semicontinuous iff✁f is upper semicontinuous.
Corollary 2.4.12 and Lemma 2.4.14are based on (HÁJEK, 1998) .
Corollary 2.4.12. Let △ be a lower semicontinuous t-norm and let ñ be its residuum.
Then for all x, y, z P r0,1s we have
x△z ↕y iff z ↕ ♣xñyq . (2.4.13) Proof. Suppose that x△z ↕y. Then
z P tz0 P r0,1s:x△z0 ↕y✉ , whence
z ↕suptz0 P r0,1s:x△z0 ↕y✉ ✏ ♣xñyq . Conversely, if z ↕ ♣xñyq then either
1. z ➔ ♣xñyq, that is
z ➔suptz0 P r0,1s:x△z0 ↕y✉ , whence z Ptz0 P r0,1s:x△z0 ↕y✉ ; or 2. z ✏ ♣xñyq, so that
z P tz P r0,1s:x△z ↕y✉ by Proposition 2.4.11.
In both cases, x△z ↕y.
The maps △and ñ satisfying (2.4.13) are called an adjoint pair.
Lemma 2.4.14. Let △: r0,1s2 Ñ r0,1s be a lower semicontinuous t-norm and let ñ be its residuum. Then, for all x, y P r0,1s,
x↕ r♣xñyq ñys .
Proof. Let x, y P r0,1s. Notice that ♣x ñ yq ✏ supA in Proposition 2.4.11, and so, x△♣xñyq ↕ y. Using commutativity of △, it is clear that
xPB :✏ tz P r0,1s:♣xñyq△z ↕y✉ . Hence,
x↕supB ✏ r♣xñyq ñys .
Lower semicontinuity of △ is a necessary hypothesis for Lemma 2.4.14 as the following example shows.
Example 2.4.15. Let ñd be the residuum of the drastic product t-norm, as defined in item 6 of Example 2.3.15. Let x✏ 2
3 and lety ✏ 1
3. Then
♣xñdyq ✏
✂2 3 ñd
1 3
✡
✏1 , as x→y. Thus,
r♣xñd yq ñd ys ✏
✒ 1ñd
1 3
✚
✏ 1 3 ➔ 2
3 ✏x , and ñd does not satisfy Lemma 2.4.14.
Lemma 2.4.16 and Prop. 2.4.17 are proved by (BRITO et al., 2018).
Lemma 2.4.16. Let X, Y ❸R, f :X ÑY be an increasing function, ♣xnq be a sequence on X such that xn ×xPX and ♣ynq be a sequence on Y.
If for all n →0, f♣xnq ↕yn, then f♣xq ↕lim inf
nÑ✽ yn↕lim sup
nÑ✽ yn .
Proof. Suppose that f♣xnq ↕yn for all n→0. Since ♣xnq is decreasing, for each n→0 we have
sup
j➙n
xj ✏xn , so that
kinf→0
✂ sup
j➙k
xj
✡
↕xn , so that we have
x✏lim sup
jÑ✽ xj ↕xn . as ♣xnq
Thus, for each n →0,
f♣xq ↕ f♣xnq ↕yn .
Now, taking the limit inferior on the right-hand side, we conclude the proof.
Proposition 2.4.17. Let △ be a lower semicontinuous t-norm, and let ñbe the residuum of △. Then for each x0, y0 P r0,1s the maps x ÞÑ ♣x ñy0q and y ÞÑ ♣x0 ñyq are upper semicontinuous.
Proof. Let x✶, y✶ P r0,1s. Let ♣xnq,♣ynqbe sequences on r0,1s converging respectively to x✶ and y✶. For each n→0, consider the following definitions:
zn♣1q✏ ♣xnñy0q ; zn♣2q ✏ ♣x0 ñynq ;
˜
x♣1q✏x✶ ; x˜♣2q ✏x0 ;
˜
x♣n1q✏ inf
m➙nxm ; x˜♣n2q ✏x0 ;
˜
y♣1q✏y0 ; y˜♣2q ✏y✶ ;
˜
yn♣1q✏y0 ; y˜n♣2q ✏ sup
m➙nym ;
˜
zn♣1q✏ x˜♣n1qñy˜n♣1q✟
; z˜n♣2q ✏ x˜♣n2q ñy˜♣n2q✟ . Notice the following:
1. The sequence♣˜x♣n1qq is constructed as the infima of the decreasing sets An ✏ txm : m➙n✉. Thus, ♣˜x♣n1qq is increasing. Furthermore asn increases the elements left in eachAn get always closer to x✶. Thus, ˜x♣n1q Õx✶. In general, ˜x♣niq Õx˜♣iq, for i✏1,2.
2. By an argument analogous to that of item 1., ˜yn♣iq ×y˜♣iq, for i✏1,2.
3. For each i✏1,2, considering monotonicity of ñin each component, one sees that for all n →0, zn♣iq ↕z˜n♣iq. Thus, lim sup
nÑ✽ zn♣iq ↕lim sup
nÑ✽ z˜n♣iq. 4. The sequence ♣˜zn♣iqqis decreasing, for i✏1,2.
In the following, i✏1,2. By Proposition2.4.11, for each n →0,
˜
x♣niq△z˜n♣iq ↕y˜n♣iq
as △ is lower semicontinuous. Forn0 →0 fixed and by monotonicity of△ we have, for each n →n0,
˜
x♣ni0q△z˜♣niq↕y˜n♣iq . Thus, an application of Lemma 2.4.16 yields
˜ x♣ni0q△
✂
lim sup
nÑ✽ z˜n♣iq
✡
↕lim inf
nÑ✽ y˜n♣iq ✏y˜♣iq . But n0 →0 is arbitrary, and so
lim inf
nÑ✽
✒
˜ x♣niq△
✂
lim sup
nÑ✽ z˜n♣iq
✡✚
↕y˜♣iq .
Remember that △ is lower semicontinuous and so, since♣˜x♣niqqconverges to ˜x♣iq, we see that
˜ x♣iq△
✂
lim sup
nÑ✽ z˜n♣iq
✡
↕y˜♣iq .
Hence we have
lim sup
nÑ✽ z˜n♣iq ↕suptzP r0,1s: ˜x♣iq△z ↕y˜♣iq✉
✏ x˜♣iqñy˜♣iq✟ . Thus,
lim sup
nÑ✽ z♣iq ↕ x˜♣iq ñy˜♣iq✟ .
For i ✏ 1 this means x ÞÑ ♣x ñ y0q is upper semicontinuous at x✶, and for i✏2,y ÞÑ ♣x0 ñyq is upper semicontinuous aty✶. Sincex✶, y✶ P r0,1s are arbitrary, both the maps xÞÑ ♣xñy0q and yÞÑ ♣x0 ñyq are upper semicontinuous.
Being able to extend classical methods to fuzzy ones is already a very powerful tool. But what would happen if what we have at hand consists in a finite subset of r0,1s, or even if the truth-values at hand are not linearly ordered? These questions are dealt with in the next section.