commutative, residuated groupoid, and its¬1-level function coincides with¬. The disconnected rotation operator, as described above, preserves associativity, conjunctivity, unit element, inte-grality, and being lattice-ordered.
For more details, see Theorem 7.22 on page 132. The proof of this theorem is a tedious but easy verification. The real achievement is to find the statement. Below we present a geometric motivation for stating Theorem 1.35.
Geometric Motivation. LetM = [0,1],¬x= 1−x, and let∗◦be an associative operation on [0,1]withx∗◦y= 0if and only ifx≤1−y. Then∗◦is conjunctive and residuated (Corollary 1.31), and the partial mappings of∗◦ are symmetric in the sense of Fig. 1.5 (see the remark before it).
This means that if we know the “upper half” of a partial mapping, then the whole partial mapping is uniquely determined. Now, if a subsemigroup of∗◦ is given on[12,1], then x∗◦ 12 = 12 for any x∈[12,1]by the conjunctive nature of∗◦. That is, thex-partial mapping at12 is equal to12. In other words, we know the “upper half” of all the x-partial mappings, and whence the subsemigroup on [12,1] uniquely determines the semigroup on [0,1]. Finally, the extension formula for ∗◦ in Theorem 1.35 can easily be computed from the pseudo-inverse property, and →∗◦ is computed from∗◦.
An MV-algebra [17, 21] is calledperfectif for every elementa, exactly one element of{a,¬a}
has finite order6. The concept of perfect algebras has been used fruitfully in the classification of MV-algebras. The idea of perfect algebras has been generalized to the much wider variety of IMTL-algebras, and it is proved that perfect IMTL-algebras are exactly the algebras obtained from a basic semihoop by disconnected rotation (for the details, see [97]). Whence, the dis-connected rotation construction provides an insight into the structure of perfect IMTL-algebras in a similar manner as the ordinal sum construction [29, 28] does into the structure of certain topological semigroups [90].
Example 1.36 The connected rotation method, which is introduced in [70], is similar to the one in Theorem 1.35 (see Theorem 7.15 on page 125). Here we only depict an example for connected rotation. Consider the product of real numbers restricted to[0,1]. Its connected rotation (with respect to¬x = −x) is depicted in Figure 1.14. Another example, the connected rotation of the minimum on[12,1], is depicted in Fig. 1.5 (top-center).
It is well known that the multiplication on the negative cone of an ordered group has a unique associative extension into the whole group (which is the group multiplication itself). In fact, the disconnected rotation construction is theuniqueassociative extension from a negatively-ordered (commutative, residuated) semigroupMintoM∗.
1.7 Applications in logic
Finally we point out briefly that our geometric description of associativity is useful not only in algebra, but in non-classical logic. Moreover, we will prove an open problem posed by three
6The order ofais the smallest powermsuch that themthpower ofais equal to0.
1 0,5
0 -0,5
-11 0 0,5
0 5 -1
-0,5 0 0,5 1
Figure 1.14: Connected rotation of the product on[0,1], see Example 1.36 leading experts of functional equations about associative functions in Chapter 3.
Therule of detachment(with respect to the implication→) is the only inference rule of the system IMTL(InvolutiveMonoidalT-norm-basedLogic) [38], defined by the following axioms:
Ax1 (ϕ→ψ)→((ψ →χ)→(ϕ→χ)), Ax2 ϕ&ψ →ψ&ϕ ,
Ax3 ϕ∧ψ →ϕ , Ax4 ϕ∧ψ →ψ∧ϕ ,
Ax5 ((ϕ→ψ)→χ)→(((ψ →ϕ)→χ)→χ), Ax6 ¬¬ϕ→ϕ .
AxEG7 ϕ&ψ →ϕ ,
AxEG8 ϕ& (ϕ→ψ)→ϕ∧ψ ,
AxEG9 (ϕ→(ψ →χ))→(ϕ&ψ →χ), AxEG10 (ϕ&ψ →χ)→(ϕ→(ψ →χ)), AxEG11 ¯0→ϕ ,
AxEG12 (ϕ→¯0)→ ¬ϕ , AxEG13 ¬ϕ→(ϕ→¯0).
1.7. APPLICATIONS IN LOGIC 43 IMTLis the logic of left-continuous t-norms having an involutive0[37]. That is, the class of left-continuous t-norms having an involutive0constitutes an algebraic semantic forIMTL. (Another algebraic semantics is the variety of IMTL-algebras.) It was proved syntactically in [49] that ax-iomsAxEG7–AxEG13can be replaced by the axioms below, and the two axiom systems determine the same logic.
AxGJ7 ϕ&ψ →ϕ∧ψ ,
AxGJ8 (ϕ&ψ →χ)→(ψ&¬χ→ ¬ϕ), AxGJ9 (ψ&¬χ→ ¬ϕ)→(ϕ&ψ →χ), AxGJ10 (ϕ→ψ)→ ¬(ϕ&¬ψ),
AxGJ11 ¬(ϕ&¬ψ)→(ϕ→ψ), AxGJ12 ϕ→ϕ& ¯1,
AxGJ13 ϕ→ ¬¬ϕ
In the second system,AxGJ8can be “intuitively understood” as ifx∗◦y≤z =⇒y∗◦ ¬z ≤ ¬x would hold for a corresponding algebraic structure, where ¬ is an involution (see Ax6 and AxGJ13). That is, this axiom captures the rotation-invariance property. (There is a similar inter-pretation forAxGJ9.) On the other hand,AxEG9andAxEG10are left out from the second system;
those axioms are usually responsible for “capturing residuation”. The basic idea of the second axiomatization is motivated by Lemma 1.27 for which a geometric explanation is given in this chapter. Another application of the rotation construction in logic is presented in [70] (Section 6).
Chapter 2
Subdomains of uniqueness
2.1 Introduction
Associative functions on real intervals were first considered by Abel in 1826 [1] and have since been studied by many other mathematicians – see the classic treaties by J. Acz´el [4] for math-ematical and historical details. A special class of associative functions, the so-called t-norms, have been applied in various mathematical disciplines including game theory, the theory of non-additive measures and integrals, the theory of measure-free conditioning, t-norm-based logics, control, preference modelling and decision analysis, and artificial intelligence since their intro-duction in 1942. They have been studied not only with their original application to probabilistic metric spaces [105], but also, in connection with semigroup theory and functional equations. For further details we refer the reader to the monograph on t-norms [78].
Many authors have focused on the identification of small subsets of the unit square which uniquely determine a continuous Archimedean t-norm ([79, 105]). The main results of such investigations are the following:
Astrictt-normT is uniquely determined by its diagonal section and the section along the graph of a strictly decreasing bijection of the unit interval. Moreover, in [8, 30] those requirements are weakened considerably. In [8] it was shown that it suffices to know the values of a strict t-norm on some appropriate subset of the two diagonals, e.g., on {(x, x)|x∈[0,1]}and on{(x,1−x)|x∈[0, ε]}, for anyε∈]0,1].
Anilpotentt-norm is uniquely determined by its diagonal section and its preimage of{0}
[16].
In the present section other subsets of the unit square are shown to admit the property that there exists a unique t-norm (either a nilpotent one or a strict one or a left-continuous one) provided that its values are given on that subset. The employed subsets are either vertical cuts of the graph of the t-normT, that is, one-place functions of the form T(., x), which can be considered as intersections of the graph of the t-norm with vertical planes, or horizontal cuts, that is, one-place functions of the formx→∗◦c(see Definition 2.3), which can be considered as limit lines of intersections of the graph of the t-norm with horizontal planes.
45
Similar investigations have been carried out in [66] for the much larger class ofleft-continuous t-norms ([63]): Certain vertical or horizontal segments of the graph of the t-norm, that is, one-place functions of the formx∗◦., and of the formx→∗◦.(→∗◦being the residuum of∗◦) have been shown to determine uniquely the left-continuous t-norm.
In Section 2.3 we shall demonstrate its plausible applicability for the subdomains of unique-ness problem.
2.2 Preliminaries
Atriangular norm(t-norm for short) is a function∗◦ : [0,1]2 → [0,1]such that for allx, y, z ∈ [0,1]the following four axioms (T1)-(T4) are satisfied:
(T1) Commutativity x∗◦y=y∗◦x
(T2) Associativity x∗◦y∗◦z =x∗◦y∗◦z
(T3) Monotonicity x∗◦y≤x∗◦z whenevery≤z (T4) Boundary condition x∗◦1 =x
(T5) Boundary condition 0∗◦y = 0
(T6) Conjunctive nature x∗◦y≤min(x, y).
It is immediate to see that (T3) and (T4) imply (T5) and that (T1), (T3) and (T4) imply (T6). A t-norms is calledcontinuousif it is continuous as a two-place function. A continuous t-norm is calledArchimedeanifx∗◦x < xholds forx∈]0,1[. A continuous Archimedean t-norm is called nilpotentif it has zero divisors (that is, if there existsx∈]0,1]such that there existsy∈]0,1]with x∗◦y= 0). A prototype of nilpotent t-norms is the so-called Łukasiewicz t-norm, given by
TL(x, y) = max(0, x+y−1).
A continuous Archimedean t-norm is called strictif it has no zero divisors. An example is the product t-norm, given by
TP(x, y) = x·y.
In fact, these are the unique examples for nilpotent and for strict t-norms up toϕ-transformation, as shown by the following theorem.
Theorem 2.1 [84]Any nilpotent t-normT is isomorphic toTL, that is, there existsϕ, which is an increasing bijection of[0,1], such thatTϕ, theϕ-transform ofT, is the Łukasiewicz t-norm.
That is,
Tϕ(x, y) :=ϕ−1(T(ϕ(x), ϕ(y))) =TL(x, y).
Any strict t-normT is isomorphic toTP, that is, there existsϕ, which is an increasing bijection of[0,1], such thatTϕ, theϕ-transform ofT, is the product t-norm. That is,
Tϕ(x, y) :=ϕ−1(T(ϕ(x), ϕ(y))) =TP(x, y).
Definition 2.1 Let[a, b] ⊂ R, a < b. Aninvolution of[a, b] is a decreasing bijection from [a, b]to[a, b]such that its composition with itself is the identity mapping of[a, b].
2.2. PRELIMINARIES 47 Definition 2.2 LetT be a t-norm, andc∈[0,1]. Thec-level setofT is defined as follows:
T−1{c}={(x, y)∈[0,1]|T(x, y) = c}
Definition 2.3 Let ∗◦ be a left-continuous t-norm. For any c ∈ [0,1] define the mapping
¬c : [0,1]→[0,1]by
¬cx= max{y ∈[0,1]|x∗◦y≤c}.
Observe that due to (T5) the set{y ∈ [0,1] |x∗◦y ≤ c}is never empty, and that the maximal element of this set always exists since ∗◦ is left-continuous and (T3) holds. ¬cx is called the residuumofxandc. We remark that the preimage of{0}, that is, the set{(x, y)∈[0,1]|x∗◦y= 0}and the function x→∗◦ 0mutually determines each other. For the properties of the residuum which are described in the forthcoming statements of this section see e.g. [43], Chap XII, page 189.)
The function¬cxis non-increasing. Byadjointness propertyandexchange propertythe following properties are understood, respectively: For anyx, y, c∈[0,1]we have
x∗◦y≤c ⇐⇒ x≤ ¬cy
¬cx∗◦y=f¬cy(x).
As well we have
¬c¬c¬cx=¬cx (2.1)
alimn↓ax→∗◦ an=x→∗◦a (2.2) wherean↓astands forlimn→∞an=a, andan> afor alln.
Proposition 2.2 For any continuous Archimedean t-norm∗◦itsc-level set and the functionfc
mutually determine each other for anyc∈[0,1]. More formally, we have T−1{c}=
{(x, fc(x))∈[0,1]|x≥c} if c∈]0,1]
{(x, y)∈[0,1]|y≤fc(x)} if c= 0 (2.3) fc(x) =
1 ifx < c
max{y|(x, y)∈T−1{c}} ifx≥c (2.4) Proof. The second row of (2.3) and the first row of (2.4) are evident by taking into account the definition of the residuum and (T4). (T4), (T5), and Bolzano’s theorem ensure the existence ofz such thatT(x, z) =c, for any0≤c≤x. In addition, we haveT(x, fc(x))≤candT(x, z)> c forz > fc(x)by the definition of the residuum. Thus, we haveT(x, fc(x)) =c. From Theorem 2.1 we conclude thatT(x, y)> T(x, z)ifT(x, z)>0andy > z. Therefore, forc >0,z < fc(x) we haveT(x, z) < c. These prove the first row of (2.3). Referring again to Bolzano’s theorem, and to (T3), the proof of the second row of (2.4) is concluded.