Applying reflection-invariance

The results of Section 1.5 provide us with a transparent tool to conjecture and prove results about so-called subdomains of uniqueness problem for left-continuous t-norms.

Definition 2.4 A subset D of [0,1]² is called a subdomain of uniqueness with respect to a class of t-normsT, if no two different t-norms∗◦₁ and∗◦₂ inT can coincide all overD. Stated it in another way, if we know the values of a t-norm∗◦ ∈ T onDthen they uniquely determines∗◦ on its whole domain[0,1]².

Notation: The notation¬^c referred to an involution in the previous sections. In what follows¬ will denote a more general mathematical object, namely a non-increasing function.

Definition 2.5 For any non-increasing function¬ of type [0,1] → [0,1]with ¬0 = 1 and

¬1 = 0denoteD^¬ ={(x, y)∈[0,1]×[0,1]|y ≥ ¬x}. That is,D^¬ is the part of the unit square which is above¬.

Theorem 2.3 Let¬be any non-increasing function of type[0,1] → [0,1]such that¬x < 1 wheneverx >0. The setD^¬is a subdomain of uniqueness with respect to the class of continuous Archimedean t-norms.

Proof. We shall prove the theorem via proving a sequence of claims. Let ∗◦ be a continuous Archimedean t-norm. Letm = inf{x∈[0,1]|x≤ ¬x, x∗◦x >0}.

Claim 1.The values of∗◦onD^¬uniquely determine the values of∗◦on[m,1]².

Proof of Claim 1. Letc∈ ]m,1]arbitrary. Since∗◦is continuous it is sufficient to prove that the values of∗◦onD^¬ uniquely determine∗◦on[c,1]². First observe[c,1]×[¬c,1]⊂D^¬since (c,¬c)∈D^¬. Due to the commutativity of∗◦we know the values of∗◦on

D0 := ([c,1]×[¬c,1])∪([¬c,1]×[c,1]). Fori∈N,i≥1denotec_i =¬c∗◦¬c∗◦. . .∗◦¬c

| {z }

i

and define

D_i :=D₀∪[c_i,1]²

Since c > 0 we have ¬c < 1. Therefore, due to the Archimedean property of ∗◦, c_i is a non-increasing sequence, and there existsr ∈N,r >1such thatcr+1 ≤c < cr.

First we shall show, by induction, that ∗◦is uniquely determined onD_r. By usingc ≤ ¬c we have (¬c,¬c) ∈ D₀; thereforeD₁ ⊂ D₀ and thus∗◦ is uniquely determined onD₁. If r= 1then the proof is ended. Next assume∗◦is uniquely determined onDi for somei < r.

Sincec < c_i+1 we have that the graph of thec_i+1-level function restricted to[c_i+1,1], that is{(x, x→∗◦c_i+1)|x∈ [c_i+1,1]}, is a subset ofD_i. It follows that∗◦is determined on the

2.3. APPLYING REFLECTION-INVARIANCE 49 part of[c_i+1,1]²which is above this graph, hence by Corollary 1.24, we conclude that∗◦is uniquely determined on[c_i+1,1]², hence onD_i+1.

Next, we show that ∗◦ is uniquely determined on [c,1]². Denote e := c→∗◦ c_r+1. Since cr→∗◦cr+1 =¬c, andc < cr+1we have¬c < e. Thus,∗◦is determined on([c, e]×[¬c, e])∪

([¬c, e]×[c, e])∪[c_r, e]². It follows that the graph of thec_r+1-level function restricted to [c, e], that is{(x, x→∗◦ c_r+1)|x ∈ [c, e]}, is a subset ofD_r, hence∗◦is determined on the part of[c, e]² which is above this graph, By Corollary 1.24, we conclude that∗◦is uniquely determined on[c, e]², and hence on[c,1]². The proof of Claim 1 is concluded.

¬

¬c c

¬c

c

c

c

c

c

c

c

c

c

. . .

c

c

Figure 2.1: Illustration: induction steps for Claim 1 in Theorem 2.3

Observe that we havem= 0if∗◦is strict. Indeed, letc∈]0,1]be arbitrary withc≤ ¬c. Suchc exists since by denotingz = limx→+0¬xwe havez >0using¬|_]0,1] 6≡ 0¹, and one can choose c∈

0,^z₂

such that ^z₂ ≤ ¬c. Observe that all positive numbersxsmaller thanchave as well the propertyx≤ ¬xsince¬is non-increasing. Hence we havem = 0 as stated, and thus the proof of Theorem 2.3 is ended for strict t-norms.

Claim 2.If∗◦is a nilpotent t-norm andtis the unique fixed point ofx7→x→_∗◦0then the values of∗◦onD^¬ uniquely determine the values of∗◦on[t,1]².

Proof of Claim 2. Ift >¬tthen[t,1]² ⊂ D^¬ and the proof is ended. Next assumet ≤ ¬t. No elementxsmaller thantsatisfiesx∗◦x >0, therefore we havem=t, and Claim 1 ends the proof of the statement.

1If¬|_]0,1]≡0then Theorem 2.3 obviously holds true.

Claim 3. If∗◦is a continuous Archimedean t-norm,c, d∈[0,1]such thatc∗◦d >0then the value c∗◦dis uniquely determined by the values of∗◦onD^¬.

Proof of Claim 3. We have already confirmed this for strict t-norms. Assume ∗◦ is nilpotent.

Denote the unique fixed point ofx 7→ x→∗◦0 byt. Letc, d ∈ [0,1]such that c∗◦d > 0.

If c = t then d > t and we arrive at Claim 2, so we can safely assume c < t < d by commutativity of ∗◦. We shall prove that the values of ∗◦ are uniquely determined on [c,1]× [d,1]. Observe that [c,1] ×[¬c,1] ⊂ D^¬ since (c,¬c) ∈ D^¬, thus due to the commutativity of∗◦and Claim 2 the values of∗◦on

D:= ([c,1]×[¬c,1])∪([¬c,1]×[c,1])∪[t,1]²

are uniquely determined. We can assume¬c > dsince otherwise(c, d)∈ Dand the proof is ended. Fori∈Ndenotec_i =t∗◦ ¬c∗◦¬c∗◦. . .∗◦¬c

| {z }

i

and define

D_i :=D∪([c_i,1]×[d,1])∪([d,1]×[c_i,1])

Since c > 0 we have ¬c < 1. Therefore, due to the Archimedean property of ∗◦, c_i is a decreasing sequence, and there existsr∈N,r >1such thatc_r+1 ≤c < c_r.

First we shall show, by induction, that∗◦is uniquely determined onD_r. We haveD₀ ⊂ D and therefore∗◦is uniquely determined onD₀, thus the proof is ended forr = 0. Assume

∗

◦is uniquely determined onD_i for somei < r. We claim that the graph of thec_i+1-level function restricted to [c_i+1,1], that is {(x, x→_∗◦ c_i+1) | x ∈ [c_i+1,1]}, is a subset of D_i. Indeed, observe thatc_i →∗◦ c_i+1 = ¬cand ¬c→∗◦ c_i+1 = c_i since∗◦ is strictly increasing on the part of the domain of∗◦where the value of ∗◦is positive on one hand, and we have c_i ∗◦ ¬c=c_i+1 > c >0on the other. By using these, an easy verification shows

(x, x→∗◦ci+1)∈















[c,1]×[¬c,1] ifx∈[c_i+1, c_i] [c_i,1]×[d,1] ifx∈[c_i, t]

[t,1]² ifx∈[t, t→_∗◦c_i+1] [d,1]×[c_i,1] ifx∈[t→∗◦c_i+1,¬c]

[¬c,1]×[c,1] ifx∈[¬c,1]

and hence the graph of the c_i+1-level function restricted to [c_i+1,1] is a subset of D_i, as stated. Referring tod > tand to Claim 2, we have that the graph of the d-level function restricted to[d,1]is a subset ofDitoo. It follows that∗◦is uniquely determined above those two level curves, and an application of Corollary 1.23 yields that∗◦is uniquely determined on[c_i+1,1]×[d,1], and hence onD_i+1.

Next, we show that∗◦is uniquely determined on[c,1]×[d,1]. Lete:=c→∗◦c_r. Since¬c= c_r→∗◦c_r+1, andc < c_rwe have¬c < e. Thus,∗◦is determined onE := ([c, e]×[¬c, e])∪ ([¬c, e]×[c, e])∪Dr. It follows that the graph of thecr+1-level function restricted to[c, e], that is{(x, x→∗◦c_r+1)|x ∈ [c, e]}, is a subset ofE, hence ∗◦is determined on the part of [c, e]² which is above the graph of thec_r+1 level function. Referring tod > tand to Claim