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3. [MATH] is a Boolean algebra, and [MATH] is the largest element of [MATH] below [MATH] for every [MATH] Proof . Let [MATH] and [MATH] ; since [MATH] is complete, [MATH] exists. We first show that [MATH] is complete, i.e. that [MATH] . Consider |
[EQUATION] It is straightforward to extend this over infinite joins of complete modal operators. . “ [MATH] ”: [MATH] is completely additive: Suppose that [MATH] , and [MATH] exists. The aim is to show that [MATH] also exists and is equal to [MATH] . In the proof we shall use the de Morgan rules for infinite sums, see ... |
[EQUATION] Since [MATH] exists, this implies that [MATH] exists and is equal to [MATH] , and therefore, [MATH] is complete. [MATH] ”: Conversely, let [MATH] ; we need to find some [MATH] such that [MATH] . The obvious candidate for [MATH] is [MATH] which exists, since [MATH] is complete. |
Since [MATH] , we have [MATH] , since [MATH] is the dual pseudocomplement of [MATH] . For [MATH] , first observe that [EQUATION] |
Thus, [EQUATION] Assume that [MATH] . Then, [MATH] , i.e. [EQUATION] If [MATH] , then [MATH] , and [MATH] , contradicting our assumption. If [MATH] , then [MATH] , and therefore, ( 4.3 ) implies that [MATH] by ( 4.3 ), a contradiction. |
: Since the open elements of [MATH] are exactly the completely additive ones by ., [MATH] is a Boolean algebra by Lemma . The join [MATH] in [MATH] coincides with [MATH] , and the meet [MATH] is given by [MATH] . Furthermore the assignment [MATH] is an interior operator by Theorem 2 of |
. This implies the second claim. The following observation comes as no surprise: Theorem 4.5 Let [MATH] be a frame; then, [MATH] |
Proof Since both [MATH] and [MATH] are completely additive, and thus determined by their action on the singletons, it suffices to show that [MATH] for every [MATH] . Let [MATH] . Then, by Lemma and a simple computation, |
[EQUATION] This proves the claim. Let w.l.o.g. [MATH] [MATH] , and suppose that [MATH] . Even though [MATH] need not be equal to [MATH] (and not even be in [MATH] ), it seems reasonable to ask whether [MATH] in [MATH] |
Lemma 4 Let [MATH] such that [MATH] , and suppose that [MATH] . Then, [MATH] implies [MATH] Proof Let [MATH] such that [MATH] and assume that there is some [MATH] such that [MATH] . Since [MATH] we have [MATH] , and since [MATH] we have [MATH] . Thus, [MATH] implies [MATH] , a contradiction. |
Corollary 1 Suppose that [MATH] and [MATH] . Then, [MATH] for all [MATH] Proof Let [MATH] [MATH] . We need to show that [MATH] . Consider |
[EQUATION] The claim now follows from Lemma setting [MATH] Example 1 The first example (from 10 , p 251] ) exhibits a modal operator on the non–complete [MATH] which has a dual pseudocomplement. Let [MATH] and let [MATH] be defined by [MATH] . Observe that [MATH] , and therefore, [MATH] does not have a finite subalgebr... |
Let [MATH] be defined by [EQUATION] Clearly, [MATH] is a modal operator. Suppose that [MATH] . If [MATH] is not an atom, then [MATH] by definition. If [MATH] , then [MATH] . Thus, [MATH] |
Let [MATH] ; our aim is to show that [MATH] . Since [MATH] , it follows that [MATH] or [MATH] . Thus, [MATH] . Let [MATH] and [MATH] . Then, [MATH] . Hence, [MATH] is the dual pseudocomplement of [MATH] |
Since [MATH] is not complete, [MATH] is not dually pseudocomplemented. It is therefore instructive to give a concrete example of a modal operator on [MATH] without dual pseudocomplement. |
Example 2 Let [MATH] , and define [MATH] by [MATH] , and [EQUATION] and extend [MATH] over [MATH] by [MATH] . Since every cofinite [MATH] contains a positive even number [MATH] and an odd number [MATH] , we note that |
[EQUATION] Furthermore, if [MATH] and [MATH] is odd, then [EQUATION] Let [MATH] be the set of positive even numbers, and [MATH] be the set of odd numbers. For [MATH] let [MATH] be the i-th nonzero even number, [MATH] , and |
[EQUATION] We extend the [MATH] additively over finite sets; then [MATH] is finite, if [MATH] is finite. Note that the [MATH] only differ in how they handle a cofinite [MATH] with [MATH] . Let [MATH] . If, say, [MATH] , then [MATH] , and [MATH] , since [MATH] . Now, |
[EQUATION] Thus, suppose that [MATH] . If [MATH] is finite, then [MATH] by ( 4.7 ) and ( 4.8 ). If both [MATH] and [MATH] are finite, then [MATH] by the definition of [MATH] . If [MATH] is cofinite, then [MATH] is cofinite, and |
[EQUATION] Thus, [EQUATION] If both [MATH] and [MATH] are cofinite, then so is [MATH] , and [MATH] . Altogether, we have shown that [MATH] |
Next, let [MATH] . If [MATH] is cofinite, then [MATH] by ( 4.4 ), and if [MATH] , then [MATH] . Let [MATH] be finite and [MATH] ; it is enough to show that [MATH] for [MATH] . By definition, |
[EQUATION] Hence, [MATH] Assume that [MATH] is a dual pseudocomplement of [MATH] . Then [MATH] for all [MATH] , and thus, [MATH] contains no positive even numbers. However, [MATH] contains all odd numbers, and thus, [MATH] by ( 4.5 ). It follows that [MATH] , a contradiction. Thus, [MATH] does not have a dual pseudocom... |
It is instructive to consider the canonical extension [MATH] of [MATH] with [MATH] . Then, [MATH] exists since [MATH] is complete, and it is equal to [MATH] by Theorem 4.5 . Let [MATH] be the principal ultrafilter of [MATH] |
[MATH] , and [MATH] be the non–principal ultrafilter of cofinite sets; furthermore, let [MATH] be the Stone embedding. Then, for [MATH] |
[EQUATION] By ( 3.2 ) and the definition of [MATH] [EQUATION] Furthermore, [EQUATION] Therefore, keeping in mind that [MATH] we obtain |
[EQUATION] This gives us [EQUATION] This shows that [MATH] for all [MATH] . On the other hand, if we define [EQUATION] then [MATH] is a minimal pair, and [MATH] for all [MATH] |
Our final example in this section exhibits a modal operator on the countable free Boolean algebra without a dual pseudocomplement. |
Example 3 Let [MATH] be the interval algebra of the rational unit interval [MATH] ; we regard [MATH] as a subalgebra of the real unit interval [MATH] . It is well known that [MATH] is the free Boolean algebra on countably many generators. In particular, [MATH] is homogenous, and every nonempty infinite open interval is... |
Let [MATH] be irrational, and [MATH] be an order isomorphism. Suppose that [MATH] and [MATH] be its canonical representation. Now, set [MATH] , and, for [MATH] |
[EQUATION] Let [MATH] ; then, [EQUATION] Thus, [MATH] . For each [MATH] let [MATH] and [MATH] ; then, [MATH] . To abbreviate notation, let [MATH] . Considering that [MATH] is an order isomorphism, we obtain |
[EQUATION] Next, let [MATH] , and [MATH] . Since [MATH] is an upper bound of [MATH] by Lemma , it follows from ( 4.11 ) that [EQUATION] |
For each [MATH] , let [MATH] , and [MATH] for all [MATH] . Then, [MATH] , and [EQUATION] the latter since [MATH] . If [MATH] , then [MATH] , and therefore, |
[EQUATION] Assume that [MATH] is a dual pseudocomplement of [MATH] . Then [MATH] for each [MATH] by ( 4.12 ); furthermore, [MATH] for each [MATH] , and thus, [MATH] , and ( 4.13 ) implies that [MATH] . Altogether, we obtain that [MATH] , a contradiction. |
As in Example , we see concretely that the pseudocomplement does not exist, because certain infinite products (or sums) do not exist in [MATH] – this time for an atomless BA. Lemma tells us that this is to be expected, since [MATH] is not complete. Furthermore, we observe that [MATH] is a closure operator on the free c... |
Decomposing discriminators It is often the case that pairs of operators are considered which, taken together, have desirable structural properties. Examples of such pairs are Galois connections or residuated mappings. In |
pairs of operators [MATH] were considered where [MATH] is a modal operator, and [MATH] is a sufficiency operator, i.e. [EQUATION] |
for all [MATH] . A weak mixed algebra (wMIA) is a structure [MATH] such that [MATH] is a modal operator, [MATH] is a sufficiency operator, and |
[EQUATION] The class of wMIAs is denoted by [MATH] . These algebras are intimately connected to algebraic models of the logic [MATH] , which was introduced by Gargov et al. . It turns out that the discriminator decomposition algebras defined below are another way of describing weak MIAs. |
Suppose that [MATH] are modal operators on [MATH] , and consider the condition [EQUATION] Clearly, [MATH] is the unary discriminator. If a pair [MATH] of modal operators satisfies ( 5.2 ) we call it a decomposing pair , and [MATH] companion of [MATH] . If [MATH] is a decomposing pair, then so is [MATH] owing to the com... |
discriminator decomposition algebra (DDA) is a bi–modal algebra [MATH] such that [MATH] is a decomposing pair. If both [MATH] and [MATH] are proper, i.e. not equal to [MATH] [MATH] is called a proper DDA . The class of DDAs is denoted by [MATH] ; by ( 5.2 ), [MATH] is a discriminator class. The relational counterpart o... |
Theorem 5.1 There is a bijective correspondence between the set of decomposing pairs and the set of pairs [MATH] such that [MATH] is a weak MIA. |
Proof Let [MATH] be a weak MIA, and set [MATH] . Then, [MATH] is a modal operator, and, for all [MATH] [EQUATION] the latter since [MATH] . Clearly, the assignment [MATH] is an injective mapping, and all that is left to show is that it is surjective. Thus, let [MATH] be modal operators such that [MATH] for all [MATH] ,... |
Thus, the classes [MATH] and [MATH] are equipollent in the sense of Tarski and Givant . As in [MATH] we are dealing with just one kind of operator instead of the two kinds in [MATH] , it is less complicated to work in [MATH] . It is also easier to apply results from the theory of modal algebras. |
Lemma 5 1. [MATH] is closed under taking subalgebras, homomorphic images and ultraproducts. 2. [MATH] is not closed under taking direct products, and thus, it is neither a variety nor a quasivariety. |
Proof 1. [MATH] is a universal class with a set of positive axioms, thus, it is closed under taking subalgebras, ultraproducts, and homomorphic images, see e.g. , Paper 5] |
2. Since [MATH] is a discriminator class, each member is simple 16 , Theorem 2.2] , and thus, [MATH] is not closed under direct products. For the rest, just note that each quasivariety is closed under direct products , Theorem 2.25] |
The equational class [MATH] was described in , Section 7] , and the results can be translated for [MATH] in a straightforward way. Given a bimodal algebra [MATH] let [MATH] be defined by |
[EQUATION] [MATH] is called a [MATH] – DDA , if [MATH] is an S5 possibility operator, i.e. if [MATH] has the following properties: |
[EQUATION] The class of [MATH] – DDAs is denoted by [MATH] ; clearly, [MATH] is an equational class. Theorem 5.2 [MATH] Proof Taking into account Theorem 5.1 , the proof is a straightforward translation of , Theorem 7.3] |
The following result relating a decomposing pair to its canonical relations comes as no surprise: Theorem 5.3 Suppose that [MATH] is a bimodal algebra. Then, [MATH] if and only if [MATH] |
Proof [MATH] ”: Let [MATH] , and assume [MATH] . Then, [MATH] and [MATH] . Thus, there are [MATH] such that [MATH] and [MATH] . Now, [MATH] , and [MATH] implies [MATH] , since [MATH] is isotone and [MATH] is a filter. Thus, [MATH] , and, similarly, [MATH] . Since [MATH] is a decomposing pair we have [MATH] . Since [MAT... |
[MATH] ”: Let [MATH] and assume that [MATH] . Then, there is an ultrafilter [MATH] such that [MATH] , i.e. [MATH] and [MATH] . It follows that [MATH] and [MATH] which contradicts [MATH] |
Theorem 5.4 If [MATH] , then [MATH] Proof This follows from the syntactic form of ( 5.2 ), see. e.g. 10 , Theorem 4.2.1] . A direct proof is as follows: Since [MATH] [MATH] . We show that [MATH] for [MATH] ; then this can be extended to all non–empty subsets of [MATH] , since the operators are isotone. Assume that [MAT... |
Proper companions Let us order [MATH] by setting [MATH] if [MATH] and [MATH] . The following observation is obvious: Lemma 6 If [MATH] is a companion of [MATH] and [MATH] , then [MATH] is a companion of [MATH] |
Theorem 6.1 Let [MATH] . If [MATH] and [MATH] are dual pseudocomplements of each other, then [MATH] is a minimal pair. If [MATH] is complete, then the converse also holds. |
Proof Since [MATH] and [MATH] are dual pseudocomplements of each other, we may suppose that [MATH] and [MATH] Let [MATH] , and [MATH] . Then, [MATH] , since [MATH] . It follows that [MATH] , since [MATH] is the dual pseudocomplement of [MATH] , hence, [MATH] . Similarly we can show that [MATH] |
If [MATH] is complete, then every [MATH] has a dual pseudocomplement, and the claim follows from the fact that the set [MATH] is dense in [MATH] |
In Example [MATH] is a minimal pair, and [MATH] for all [MATH] . This shows that the assumption of completeness in the [MATH] direction cannot be removed. |
A companion [MATH] of [MATH] is called proper , if [MATH] . Note that this notion is not symmetric: If [MATH] , then every [MATH] is a proper companion of [MATH] , but [MATH] is not a proper companion of any [MATH] . A discriminating pair [MATH] is called proper if both [MATH] |
It turns out that the property of [MATH] having a proper companion is a [MATH] first order property, as the following result shows: |
Theorem 6.2 [MATH] has a proper companion if and only if [EQUATION] Proof [MATH] ”: Suppose that ( 6.1 ) is not true, i.e. [EQUATION] |
For [MATH] , set [MATH] . If [MATH] is an upper bound of [MATH] , then [MATH] for all [MATH] , i.e. [MATH] . It now follows from ( 6.2 ) that [MATH] , i.e. [MATH] , hence, [MATH] for all [MATH] . By Lemma [MATH] has a dual pseudocomplement [MATH] , and [MATH] . Therefore, [MATH] which is the smallest companion of [MATH... |
[MATH] ”: Let [MATH] witness ( 6.1 ); in particular, [MATH] . We shall consider two cases: 1. [MATH] : By the hypothesis, [MATH] . Set |
[EQUATION] Clearly, [MATH] . Let [MATH] . If [MATH] , then [MATH] , and if [MATH] , then [MATH] . It follows that [MATH] is a proper companion of [MATH] |
2. [MATH] : Define [MATH] by [EQUATION] Clearly, [MATH] , and [MATH] since [MATH] . Let [MATH] . If [MATH] , then [MATH] by the hypothesis, and [MATH] , which implies [MATH] . If [MATH] , then [MATH] . Altogether, [MATH] is a proper companion of [MATH] |
This completes the proof. Since ( 6.1 ) holds if and only if [MATH] has a nonzero lower bound for some [MATH] , we obtain Corollary 2 |
The following statements are equivalent: [EQUATION] Since [MATH] , the non–existence of a proper companion is in some sense an expression of continuity of [MATH] at [MATH] . One may also interpret this as completeness of [MATH] at [MATH] |
Corollary 3 If [MATH] is an atom of [MATH] with [MATH] , then [MATH] has a proper companion. Proof Set [MATH] , and [MATH] ; then, [MATH] and [MATH] are witnesses for ( 6.1 ). |
Theorem 6.3 If [MATH] is subdirectly irreducible, and [MATH] is a closure operator, then [MATH] has a proper companion. Proof Considering [MATH] and [MATH] we see that ( 6.1 ) is equivalent to |
[EQUATION] Since [MATH] is subdirectly irreducible and [MATH] is a closure operator, we can use Rautenberg’s criterion in the form ( 3.5 ) to obtain some [MATH] such that [MATH] for all [MATH] . Setting [MATH] and [MATH] in ( 6.8 ) gives the desired result. |
The following examples shed light on the connection between [MATH] having a proper companion and [MATH] being dense. Example 4 Suppose that [MATH] is atomless and [MATH] is a dense ideal of [MATH] . Define |
[EQUATION] Clearly, [MATH] is a closure operator. Let [MATH] be a companion of [MATH] . If [MATH] , then, since [MATH] is atomless, there are [MATH] such that [MATH] and [MATH] . By ( 5.2 ), [MATH] , and therefore, [MATH] and [MATH] . Since [MATH] , we have [MATH] , and it follows that |
[EQUATION] If [MATH] , there is some [MATH] such that [MATH] , since [MATH] is dense. Since [MATH] and [MATH] , we have [MATH] . Hence, [MATH] is not proper. |
The next example destroys density of [MATH] while keeping part of the previous example. Recall that [MATH] is called homogeneous if [MATH] for every [MATH] |
12 , Definition 9.12.] . In particular, every infinite free algebra is homogeneous. Example 5 Suppose that [MATH] is homogeneous, [MATH] , and that [MATH] is an isomorphism. Define [MATH] by |
[EQUATION] Then, [MATH] , and thus, [MATH] is not dense in [MATH] . Since [MATH] , we see that [MATH] is normal: Let [MATH] . Then, |
[EQUATION] Clearly, [MATH] . Furthermore, [EQUATION] and therefore, [EQUATION] Thus, [MATH] is a closure operator. Let [MATH] ; by Corollary it is sufficient to show that [MATH] . In what follows, we suppose that an infinite product [MATH] is taken in the completion [MATH] of [MATH] . If [MATH] , then [MATH] in [MATH] |
[EQUATION] the latter since [MATH] is atomless. Our final example shows that density of [MATH] is in general not sufficient to show that [MATH] has no proper companion. |
Example 6 Let [MATH] be homogeneous, and [MATH] . Furthermore, let [MATH] and [MATH] . Since [MATH] and [MATH] , there are isomorphisms [MATH] , and [MATH] . Furthermore, [MATH] implies that [MATH] . Define [MATH] as follows: |
[EQUATION] Then, [MATH] is well defined since [MATH] . Furthermore, [MATH] if [MATH] , and [MATH] if [MATH] Since [MATH] [MATH] is normal. Let [MATH] . If, say, [MATH] , then [MATH] , and thus, [MATH] . Since [MATH] , we also have [MATH] |
Let [MATH] . Then, [EQUATION] Thus, [MATH] Let [MATH] . If [MATH] , then [MATH] for some [MATH] . If [MATH] , then there is some [MATH] such that [MATH] . Hence, [MATH] is dense in [MATH] |
Let [MATH] be defined by [EQUATION] Clearly, [MATH] . Let [MATH] . If [MATH] , then [MATH] , and if [MATH] , then [MATH] . Hence, [MATH] is a proper companion of [MATH] |
Thus, we observe that 1. There is some [MATH] such that [MATH] is dense and [MATH] does not have a proper companion (Example ). 2. |
There is some [MATH] such that [MATH] is a closure operator, [MATH] is not dense and [MATH] has no proper companion (Example ). 3. |
There is some [MATH] such that [MATH] is dense and [MATH] has a proper companion (Example ). Therefore, the properties of [MATH] dense and [MATH] having a proper companion are independent. However, if [MATH] has additional properties, the situation is different, as we shall show below. |
For a modal operator [MATH] on [MATH] , its [MATH] th iteration is defined as usual: For [MATH] [MATH] [EQUATION] In analogy to the corresponding property of frame relations, we say that a modal operator [MATH] is [MATH] –transitive if, for all [MATH] |
[MATH] Theorem 6.4 Suppose that [MATH] is atomless, and [MATH] is [MATH] -transitive for some [MATH] . If [MATH] is dense in [MATH] , then [MATH] does not have a proper companion. |
Proof Assume that [MATH] has a proper companion. By Theorem 6.2 , there are [MATH] such that [MATH] implies [MATH] . Let [MATH] such that [MATH] ; such [MATH] exists, since [MATH] is dense by the hypothesis. By ( 6.1 ), [MATH] , and thus, [MATH] , since [MATH] is [MATH] -transitive. Note that this is not possible in Ex... |
Again by density there is some [MATH] such that [MATH] ; since [MATH] is atomless, we may suppose that [MATH] . But then, [MATH] implies [MATH] , a contradiction. |
Example shows that the converse is not true, thus, density of [MATH] in [MATH] is too strong a property to follow from not having a proper companion, even if [MATH] is a closure operator. |
The modal axiom [MATH] can be [MATH] -generalized in two ways, see e.g. , Ch. 4.3, pp. 136f] [MATH] B() [MATH] Corollary 4 The possibility operator of the Lindenbaum–Tarski algebra [MATH] of the logics [MATH] and [MATH] does not have a proper companion for every [MATH] |
Proof Since [MATH] is the countable free Boolean algebra, it is atomless. Furthermore, says that [MATH] is [MATH] –transitive, and and B() imply that [MATH] is dense in [MATH] |
Therefore, the modal operator of the countable Lindenbaum–Tarski algebra of an axiomatic extension of K4B – in particular, that of S5 – does not have a proper companion. On the other hand, Corollary shows that any nontrivial modal operator on an algebra with at least one atom has a proper companion, in particular, that... |
# Source: arxiv 1805.11909 # Title: Quantitative approach to multifractality induced by correlations and broad distribution of data # Sections: all # Downloaded: 2026-03-03T04:47:22.228443+00:00 |
Quantitative approach to multifractality induced by correlations and broad distribution of data Abstract We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach dif... |
Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, |
31-342 Kraków, Poland Institute of Theoretical Physics, University of Wrocław, Pl. M.Borna 9, 50-204 Wrocław, Poland [EQUATION] Keywords : multifractality, spurious multifractality, time series analysis, autocorrelations, symmetric and asymmetric distributions, multifractal detrended analysis, generalized Hurst exponen... |
PACS: 05.45.Tp, 89.75.Da, 05.40.-a, 89.75.-k, 89.65.Gh corresponding authors; e-mail: darusz.grech@uwr.edu.pl Introduction and motivation |
Multifractality is commonly considered as very interesting feature of complex and composite systems which attracts a lot of attention in many areas of science. The multifractal properties of time series are extensively studied because of their omnipresence in various phenomena in nature connected with complexity like t... |
, astronomy , climate phenomena physiology , text structure , physics or finances . This fragmentary list is far from being exhaustive and does not cover enormous number of publications on the subject. |
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