10.5 Finite Involutive FL e -chains
10.5.2 Finite involutive FL e -chains with positive rank
As a corollary of Theorem 10.10, it is sufficient to investigate finite involutive FLe-chains with positive rank only. Indeed, any theorem about a positive rank algebra can readily been trans-formed into a corresponding theorem about a non-positive rank algebra by applying the bijection mentioned above. We will refer to this fact asfinite skew dualityin the sequel.
3Note that¬is the involution of then+ 1-element chain and not the originaln-element chain.
10.5. FINITE INVOLUTIVE FLE-CHAINS 169 We will characterize some subclasses of finite involutive uninorm chains in terms of their under-lying t-norm and t-conorm operations based of the notion of rank. The ‘if’ part of each character-ization can be shown by giving a method of constructing finite involutive uninorm chains with a given rank, while the ‘only if’ part states that this construction covers all finite involutive uninorm chains with the given rank.
The smallest possible positive rank is1. The next theorem treats this case.
Theorem 10.11 (rank 0, rank 1) Letn ≥ 1. Then, ∗◦is a finite involutive uninorm on the chain{1,2, . . . , n}with rank0(resp. rank1) iffnis odd (resp. nis even) and
x∗◦y=
min(x, y) ifx≤ ¬y
max(x, y), ifx >¬y . (10.15) Proof. Denote the underlying universe of the algebra by {1,2, . . . , n}. If the rank is 0, that is, t = f, then (10.13) shows that n is odd, and by denotingk = n+12 we havet = f = k. If the rank is1, that is,t =f + 1, then (10.13) shows thatnis odd, and by denotingk = n+22 we have f =k−1,t=k.
The theorem holds obviously when eithern= 1orn= 2. So we may safely assumen ≥3. Note thatk ≥2in this case. We have
1∗◦ {1,2, . . . , n}= 1 and n∗◦ {k, k+ 1, . . . , n}=n. (10.16) by (10.4) and (10.5), respectively.
Next we show 2∗◦2 = 2. Referring to item3in Proposition 10.9, suppose otherwise. That is, 2 ∗◦2 = 1. Then we have 2→∗◦1 ≥ 2 and hence 2→∗◦1 ≥ ¬(2→∗◦1) by (10.8). Thus we can infer 2→∗◦1 ≥ k = t which together with the monotonicity of ∗◦ and (10.6) implies 1 = (2→∗◦1)∗◦ ¬(2→∗◦1)≥t∗◦ ¬(2→∗◦1) =¬(2→∗◦1). This implies2→∗◦1 = n, that is2∗◦n = 1which contradicts to2∗◦n ≥2∗◦t = 2.
Referring to item4in Proposition 10.9, we can derive a contradiction either by using induction onn or by assuming a minimal counterexample for (10.15). Finally, an easy verification shows that the operation given in (10.15) is an involutive uninorm algebra, hence the proof is completed.
According to finite skew duality it is enough to prove Theorem 10.11 only for rank 1since an application of Theorem 10.10 for the rank1case immediately yields also Theorem 10.11 for rank 0.
Corollary 10.12 IULplust↔f does not have the finite model property.
Proof. It has been shown in [88] that IULis complete with respect to the class of involute FLe-chains. Idempotency is easily falsified by taking a suitable involutive uninorm on[0,1]which satisfiest=f. For example one can take the involutive uninorm at item1in Theorem 10.8. We remark that this uninorm can be considered as the symmetrization of the dual of the product t-norm (cf. [68]), and as well as a representative of the class of representable unit-norms (see [36]
where representable uninorms are called aggregative operators). On the other hand, it follows from Theorem 10.11 that all finite involutive uninorms obey idempotency.
Definition 10.9 A finite involutive FLe-algebrah{1,2, . . . , n},∗◦,≤, t, fiis >⊥-indecompos-able if it has no subalgebra on[2, . . . , n−1].
Remark 10.13 Observe that the solving of Q2 for finite chains requires only to investigate
>⊥-indecomposable algebras: To show it, consider a finite involutive uninorm∗◦on{1,2, . . . , n}
with a positive rank. If∗◦is not>⊥-indecomposable then it has a subalgebra on{2, . . . , n−1}
and thus we have 2→∗◦(n−1) ≤ n − 1 which implies 2∗◦ n = n. Therefore, in addition to (10.16), n ∗◦ {2,3, . . . , n} holds also, that is, the Cayley table of ∗◦ is uniquely determined by the Cayley table of the subalgebra, which is either >⊥-indecomposable or has a subalgebra on {3, . . . , n−2}, etc.
Denotethe drastic t-norm on{1,2, . . . , n}by xy=
1 ifx, y < n
min(x, y) otherwise . (10.17)
Theorem 10.14 (rank 2)Let n ≥ 3odd. Then, ∗◦is a>⊥-indecomposable finite involutive uninorm on the chain {1,2, . . . , n} with rank 2 if and only if its underlying norm (resp. t-conorm) ison the n+32 -element chain (resp. an arbitrary t-conorm on the n−12 -element chain).
Proof.
1. Denote∗◦the monoidal operation of an involutive uninorm on{1, . . . , n}with rank2. Item 2in Proposition 10.9 shows thatn is odd, and usingt = f + 2 and by denotingk = n+32 we obtainf = k −2and t = k. Since the rank is2, we haven ≥ 3and thusk ≥ 3. If 2∗◦2 = 2then item4in Proposition 10.9 shows that the algebra is not>⊥-indecomposable.
Therefore, by item 3in Proposition 10.9, we have2∗◦2 = 1.This yields 2 ≤ 2→∗◦1and thus we obtain thatt−1, which is the fixed point of¬, is less than or equal to2→∗◦1by (10.8). In fact, we have2→∗◦1 =t−1since2∗◦t = 2. Hence we have(t−1)∗◦(t−1) = (t−1)∗◦¬(t−1) = 1by (10.6). By the monotonicity of∗◦we have[1, t−1]∗◦[1, t−1] = 1 and thus the proof of1. is concluded.
2. Denote∗◦the monoidal operation ofU⊕. Since the rank is2we havet = n+32 andf = n−12 by (10.13). Here, we will verify only the associativity of∗◦, as the rest is immediate.
Referring to (10.2) and (10.17), for any1< x < t, we have
x∗◦n =¬(x→∗◦1) =¬(t−1) = f+ 1. (10.18) Therefore, by monotonicity, we havex, y ≥tand hencex∗◦y =x⊕ywheneverx∗◦y≥t.
This, together with the associativity of⊕yields
(x∗◦y)∗◦z =x∗◦(y∗◦z) (10.19) when either(x∗◦y)∗◦z ≥torx∗◦(y∗◦z)≥t.
So, we are going to prove (10.19) holds when either(x∗◦y)∗◦z ≤f orx∗◦(y∗◦z)≤f.
10.5. FINITE INVOLUTIVE FLE-CHAINS 171 By item5in Proposition 10.5 we have
(x∗◦y)∗◦z ≤f if and only if x∗◦(y∗◦z)≤f. (10.20) Now assume(x∗◦y)∗◦z ≤f and consider the following three cases:
• Assume x, y ≥ t and z ≤ t. Since (x∗◦y) ∗◦ z ≤ f, we have (x∗◦y) ∗◦ z =
¬(x⊕y→⊕¬z)by (10.12). By (10.20) we havex∗◦(y∗◦z) ≤ f and hence we have x∗◦ (y∗◦z) = ¬(x→⊕¬(y∗◦z)) by (10.12). Referring to x ∗◦ (y∗◦z) ≤ f again, we have x ≤ ¬(y∗◦z) and hence we have t ≤ ¬(y∗◦z). Therefore, we obtain y∗◦z ≤ ¬t = f and hence y∗◦ z = ¬(y→⊕¬z) holds by (10.12). Consequently,
¬(x→⊕¬(y∗◦z)) = ¬(x→⊕¬(¬(y→⊕¬z))) = ¬(x→⊕(y→⊕¬z)) holds, and thus (x∗◦y)∗◦z andx∗◦(y∗◦z)are clearly equal.
• Assumex, z ≥ tandy ≤t. Then we have(x∗◦y)∗◦z =z∗◦(x∗◦y) = (z∗◦x)∗◦y= (x∗◦z)∗◦y = x∗◦(z∗◦y) = x∗◦(y∗◦z) by using commutativity and the result in the previous case.
• Assumex ≥ t andy, z ≤ t. If any of yorz is equal to tthen (x∗◦y)∗◦z is clearly equal tox∗◦(y∗◦z); so we may assumey, z < e. Sincex ≥tandy < ewe have that x∗◦y≤n∗◦y =f+ 1< eby (10.18). Therefore, we have(x∗◦y)∗◦z ≤(f + 1)∗◦z = (f + 1)z = 1. On the other hand, we havex∗◦(y∗◦z) =x∗◦(yz) =x∗◦1 = 1 by (10.4).
• Ifx, y, z ≤tthen both(x∗◦y)∗◦zandx∗◦(y∗◦z)are equal toxyz.
By Proposition 10.1, (10.19) follows from them. Summing up what we obtained that (10.19) holds if either (x∗◦y) ∗◦ z or x∗◦ (y∗◦z) is in [1, f] ∪ [t, n]. Therefore (10.19) holds as well if(x∗◦y)∗◦z =t−1.
Corollary 10.15 LetCnbe the number of conorm operations on ann-element chain. Then the number of>⊥-indecomposable involutive uninorms on an n-element chain with rank 2 equals to Cn−1
2 . Also the number of involutive uninorms on an n-element chain with rank 2 equals to
n−1 2
X
i=1
Ci.
Proof. The first statement is a straightforward consequence of Theorem 10.14, the second follows by an easy induction whose induction step is described in Remark 10.13.
In the spirit of finite skew duality, from Theorem 10.14 we obtain the characterization for the−1 rank case:
Corollary 10.16 (rank -1) Let n be an even number such that n ≥ 4. Then, ◦ is a finite involutive uninorm on the chain{1,2, . . . , n}with the rank−1satisfying(n−1)◦(n−1) =nif and only if its underlying t-norm (resp. t-conorm) is a t-norm⊗on the n2-element chain satisfying 2⊗2 = 2(resp. the dual ofon the n+22 -element chain).
Proof. First observe that>⊥-indecomposability of a uninorm∗◦of a positive rank on a chain with at least three elements is equivalent to the condition that2∗◦2 = 1by item4in Proposition 10.9.
Thus the condition(n−1)◦(n−1) = nfollows from the skew duality (see Theorem 10.10.i).
Denotem = n−1, and apply Theorem 10.10 for the m-element chain in Theorem 10.14. We havem∗◦m =mby (10.5), and hence the construction ofU∇in Definition 10.8 ensures2◦2 = 2⊗2 = 2. On the other hand,2◦2 = 2must hold by the item5in Proposition 10.9.
The biggest possible rank on the n-element chain isn −1; in fact this happens when the unit elementtis equal tonand thusf is equal to1. It is straightforward to verify that algebras of the rankn−1coincide with Girard monoids on finite chains. Using the twin-rotation construction it can be reformulated as follows:
Proposition 10.17 (rank n-1)Letn ≥1. Then,∗◦is a finite involutive uninorm on the chain {1,2, . . . , n} with rank n−1 if and only if its underlying t-norm (resp. t-conorm) is a Girard monoid on then-element chain (resp. the t-conorm on the one-element chain).
Again, in the spirit of the finite skew duality, from Proposition 10.17 we obtain the characteriza-tion for the smallest possible rank:
Corollary 10.18 (rank 3-n) Letn ≥ 2. Then, ∗◦ is a finite involutive uninorm on the chain {1,2, . . . , n}with rank3−nif and only if its underlying t-norm (resp. t-conorm) is the (unique) t-norm, namely, the minimum, on the two-element chain (resp. the dual of any Girard monoid on then−1-element chain).
Proof. Denote m = n−1, and apply Theorem 10.10 for the m-element chain in Proposi-tion 10.17.
Theorem 10.19 (rank n-3) Letn ≥ 3. Then, ∗◦ is a finite involutive uninorm on the chain {1,2, . . . , n}with rankn−3if and only if its underlying t-norm satisfies condition 1 in Defini-tion 10.4 and its underlying t-conorm coincide with the maximum operaDefini-tion on the two-element chain.
Proof. Since the rank of∗◦ isn−3it follows thatt = n−1andf = 2. Sincet = n−1, it follows that the universe of the underlying t-norm and t-conorm has n −1 and 2 elements, respectively. There is only one t-norm on the two-element chain, namely the minimum operation, hence the underlying conorm of∗◦should be equal to it.
Thus, by Proposition 10.5, it remains to prove that the twin-rotation ∗◦of ⊗ and ⊕is asso-ciative, where⊗is any t-norm on ann−1-element chain which satisfies condition 1 of Defini-tion 10.4, and⊗is the minimum operation on the two-element chain. To this end, assume (10.12) with X1 = {1,2, . . . , n−1} andX2 = {n−1, n} and also with ⊗and ⊕as above. We will frequently use (10.12) in the rest of the proof without explicitly referring to it. Note that we have n∗◦1 = ¬(n→⊕¬1) =¬(n→⊕n) = ¬n= 1and thus, forx∈ {1,2, . . . , n}we have
x∗◦1 = 1. (10.21)
10.5. FINITE INVOLUTIVE FLE-CHAINS 173 Referring to Proposition 10.1 it is enough to check
(x∗◦y)∗◦z =x∗◦(y∗◦z) (10.22) formax(x, y, z) = z. Therefore, we can safely assume z = n. In fact, ifz < nthen we have x, y < n, and using the fact that∗◦coincides with⊗on{1,2, . . . n−1}we obtain(x∗◦y)∗◦z = (x⊗y)⊗z =x⊗(y⊗z) =x∗◦(y∗◦z), as required. In addition we may assumex, y >1since by referring to (10.21), both sides of (10.22) are equal to1if any ofxandyis1.
i. Ifx=y=nor ifx=nandy < nthen (10.22) clearly holds true.
ii. Assume x < n and y = n. Referring to item 4 in Proposition 10.5 twice we have (x∗◦n)∗◦n = ¬(x∗◦n→∗◦¬n) = ¬(x∗◦n→∗◦1) = ¬(¬(x→∗◦1)→∗◦1), whereas by (10.5) we havex∗◦(n∗◦n) = x∗◦n =¬(x→∗◦1). Thus we need to verify¬(x→∗◦1)→∗◦1 = x→∗◦1.
By monotonicity we have ¬(x→∗◦1)→∗◦1 = ((x→∗◦1)→∗◦2)→∗◦1 ≤ ((x→∗◦2)→∗◦2)→∗◦1 = x→∗◦1 so it remains to prove ¬(x→∗◦1)→∗◦1 ≥ x→∗◦1. By the adjointness property it is equivalent to(x→∗◦1)∗◦ ¬(x→∗◦1)≤1which holds by (10.6).
iii. Assumex, y < n. Then(x∗◦y)∗◦n =¬(x∗◦y→⊗¬n) = ¬(x∗◦y→⊗1) = ¬(x⊗y→⊗1), whereas x ∗◦ (y∗◦n) = x ∗◦ ¬(y→⊗1). Now if ¬(y→⊗1) = n then x ∗◦ ¬(y→⊗1) =
¬(x→⊗¬(¬(y→⊗1))) =¬(x→⊗(y→⊗1))and so the two sides are equal. If¬(y→⊗1)<
nthenx∗◦¬(y→⊗1) =x⊗¬(y→⊗1)so it suffices to provex⊗¬(y→⊗1) =¬(x⊗y→⊗1).
Now, observe that we havex⊗ ¬(y→⊗1) = ¬((x⊗ ¬(y→⊗1))→⊗f) which is equal to
¬(x→⊗(¬(y→⊗1)→⊗f)) = ¬(x→⊗(y→⊗1))which in turn is equal to¬(x⊗y→⊗1)and the proof is ended.
Corollary 10.20 (rank 5-n) Letn ≥ 5. Then, ∗◦ is a finite involutive uninorm on the chain {1,2, . . . , n} with rank 5− n if and only if its underlying t-norm coincide with the minimum operation on the three-element chain, and its underlying t-conorm satisfies condition 2 in Defini-tion 10.4.
Proof. Denote m = n − 1, and apply Theorem 10.10 for the m-element chain in Theo-rem 10.19. It is immediate from the construction that the underlying t-conorm coincide with the minimum operation on the three-element chain. An immediate consequence of Theorem 10.10 is that the underlying t-norm of a positive rang algebra satisfies conditioniin Definition 10.8 if and only if the underlying t-conorm of its skew dual satisfies conditioniiin Definition 10.8.
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