Sec-tion 1.2.2. To prove the other direcSec-tion, assume, by contradicSec-tion, that there exists a c ∈ M such that the ? operation ofMc is not associative. That is, there exist x, y, z ∈ Mc such that [x]c?([y]c?[z]c) 6= ([x]c?[y]c)?[z]c. Mcis isomorphic to the quantic quotient with respect to
cand hence we havexc∗◦(yc∗◦zc)6= (xc∗◦yc)∗◦zc, which contradicts the associativity of∗◦.
1.⇐⇒3. Assume ∗◦ is associative. Then, for any c ∈ M, Mc is a semigroup and hence by the Rotation Invariance Lemma the ? operation in Mc is rotation-invariant with respect to the level function defined by the least element of Mc. To prove the other direction, assume, by contradiction, that there exists x, y, z ∈ M such that (x∗◦y)∗◦ z = c 6= d = x∗◦ (y∗◦z).
Then we have either [c]c 6= [d]c or [c]d 6= [d]d. Indeed, if both were equalities then, by using basic properties of closure operators, we would haved ≤ dc = cc = c andc ≤ cd = dd = d, a contradiction. Hence we may safely assume [c]c 6= [d]c. Then the following holds in Mc: ([x]c?[y]c)?[z]c= [c]c6= [d]c= [x]c?([y]c?[z]c), which contradicts the associativity of?.
Remark 1.12 (Geometric Interpretation III.) It follows from Propositions 1.1 and 1.11 that the checking of the associativity of a commutative, residuated operation ∗◦ amounts to the verifying of the rotation-invariance property with respect to the involution defined by the least element in allc-quotients of it. In this way, using the notion of quantic quotients, the general case can be handled with the help of the involutive case:
Example 1.13 Consider the operation which is depicted on the top-left of Fig 1.4. As an application of Section 1.3, we shall show that it is not associative.
Indeed, the horizontal cut of the graph at 25, and the 25-level function (that is, x →∗◦ 2 5) is depicted in Fig 1.4 (top-middle and top-right, respectively). Its range (that is the set of 25-closed elements) is 2
5 by 45 in the 25-quotient. That is, in the sequel we shall consider the original operation restricted to2
5,45
(bottom-left). By Proposition 1.11/3, this operation has to be rotation-invariant with respect to its least element, which is 25. But the 25-level function of the operation is a “straight line”, and therefore by Lemma 1.4 the (algebraic) rotation-invariance property in question has to appear as an invariance of its graph with respect to a real (geometric) rotation of 2
5,453
. Having a look at Fig. 1.4 (bottom-right), one can immediately see that is it not the case, whence the original operation is not associative.
1.4 Geometry of associativity in 2 dimensions
The aim of the present section is to point out that associativity of commutative operations can be seen even from the sectionsof the three-dimensional graph. Under sections we mean one-place functions of the form· ∗◦x, and¬c., which aretwo-dimensionalobjects.
Definition 1.3 Define the pseudo-inverse of antitone mappings as follows: Let (M,0)be a poset with least element 0, and H be a poset. Further, letf : M → H be an antitone mapping
10,8
Figure 1.4: The operation, depicted on the top-left isnotassociative, see Example 1.13 such that for all y ∈ H, the least upper bound of the set {t ∈ M | f(t) ≥ y} exists inM. By declaringsup∅= 0, letf(−1) :H →M be a function defined by
f(−1)(y) = sup{t∈M |f(t)≥y}.
Callf(−1)the pseudo-inverse off. This definition is a particular case of a straight generalization of a concept (called quasi-inverse) for real functions [105, 70]. Iffis an order reversing bijection thenf(−1), of course, coincides with the usual inverse off.
Remark 1.14 The notion of pseudo-inverses of monotone functions on intervals ofRhas a geometric interpretation: There is a simple geometric way to construct the graph of the pseudo-inversef(−1)from the graph off [105].
i. Draw vertical line segments at discontinuities off.
ii. Reflect the graph off at the first median, i.e., at the graph of the identity function.
iii. Remove any vertical line segments from the reflected graph except for their upmost points.
1.4. GEOMETRY OF ASSOCIATIVITY IN 2 DIMENSIONS 25 The intuitive idea of this section is to characterize associativity by following equality (compare with (1.5))
c y
x = c xy. Therefore, the crucial definition of the section is
Definition 1.4 Let(M,≤)be a poset and letM= (M,∗◦,→∗◦)be a commutative residuated groupoid. We say that ∗◦ admits the pseudo-inverse property with respect to c ∈ M if for all x, y, z ∈M we have
x→∗◦¬cy=¬c(x∗◦y). (1.5)
Even though the pseudo-inverse property is an algebraic notion, it has a strong connection to pseudo-inverses of monotone functions as we shall shortly see. For anyc∈M, we have that the rotation invariance property with respect to¬cand the pseudo-inverse property with respect toc are equivalent under the assumption of commutativity. Moreover, those are equivalent to a kind of symmetry of certain one-place mappings. The next statement extends the Rotation Invariance Lemma and thus provides other characterizations for the associativity of commutative operations.
Lemma 1.15 (Pseudo-Inverse Property Lemma)Assume the hypothesis of Lemma 1.3. Let c∈M. The following statements are equivalent:
1. ∗◦is rotation-invariant with respect to¬c,
2. ∗◦has the pseudo-inverse property with respect toc,
3. For anyx∈M, thec-complement of the vertical section of∗◦atxdefined byfx :M →M, y7→ ¬c(x∗◦y)
is the pseudo-inverse of itself.
Proof. First we shall prove the equivalence between1and2. By residuation,x∗◦y≤ ¬czholds if and only ifx→∗◦¬cz ≥y. By the pseudo-inverse property, it is equivalent to¬c(x∗◦z)≥y. This is equivalent tox∗◦z ≤ ¬cyby using adjointness, commutativity, and adjointness, respectively, and it is equivalent toz ∗◦x ≤ ¬cyby commutativity. Finally, it is equivalent tox∗◦y ≤ ¬cz by the rotation-invariance of∗◦. This ends the proof2.
Next, we prove the equivalence between2and3. It is straightforward to see thatfxis antitone.
We claim that least upper bound of {t ∈ M | fx(t) ≥ y} exists for all x, y ∈ M. Indeed, {t∈M |fx(t)≥y}={t∈M | ¬c(x∗◦t)≥y}={t∈M |x∗◦t≤ ¬cy}, and the supremum of this set exists sinceM is residuated and the supremum is the existing greatest element of the set.
We havefx(y) =¬c(x∗◦y) = x→∗◦¬cy= sup{t∈M |x∗◦t≤ ¬cy}= sup{t ∈M| ¬c(x∗◦t)≥ y}= sup{t ∈M |f(t)≥y}, again, if and only if the pseudo-inverse property holds.
2See the remark after the proof of Lemma 1.3.
That is, the pseudo-inverse property admits the following geometric interpretation: Let M be linearly ordered, e.g., M = [0,1]. Lemma 1.15 shows that for any x, c ∈ [0,1] the graph of vx,c : [0,1] → [0,1], y 7→ ¬c(x∗◦y) has the following geometric property: First, extend its discontinuities with vertical line segments. Then the graph obtained is invariant under the reflection at the line given byy =x.
The case when the easiest to see the geometric property above is if ¬cis a kind of involution of[c,1], which is a “straight line”, that is, when¬cx = 1 +c−x, x ∈ [c,1]. Then the vertical
Figure 1.5: Graphs of monoids on[0,1]and their vertical cuts at0.5
Geometric Motivation for Lemma 1.15/1=⇒3.
Horizontal cuts of the graph of ∗◦are curves that are symmetric in the sense of Remark 1.14 due to commutativity of∗◦. Assume ¬0x = 1−x, and assume that∗◦ is rotation invariant with respect to¬0. The image of a horizontal cut under ρ(defined in Lemma 1.4) is part of a partial mapping. Sinceσmaps symmetric curves into symmetric curves, the kind of symmetry which is described in Lemma 1.14 is preserved for the partial mappings. Thus,fx :M →M,y7→1−x∗◦y is the pseudo-inverse of itself (see Fig. 1.6).