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Proof of Claim 3. Let [MATH] be given by [MATH] . By definition of [MATH] [MATH] is injective. It is also onto since [MATH] and hence for every [MATH] , we can choose [MATH] and [MATH] . It remains to see that [MATH] is a group homomorphism. Let [MATH] , and take [MATH] as in Claim 2(1). We then have |
[EQUATION] as required. The ‘moreover’ clause is clear by the definitions and Fact 4.3 5. The proof of Theorem 1.1 We are now ready to prove Theorem 1.1 . The left-to-right direction is immediate, since every [MATH] -definable group is strongly large. For the right-to-left direction, we prove that any strongly large gr... |
Theorem 5.1 Let [MATH] be a strongly large group with [MATH] and [MATH] . Then there are L-definable sets [MATH] and [MATH] , with [MATH] |
a definable injective map [MATH] , with [MATH] , and an [MATH] -definable map [MATH] , such that for every [MATH] [EQUATION] The rest of this section is devoted to proving Theorem 5.1 . The proof runs through the five first steps mentioned in the introduction. For [MATH] , we write [MATH] for [MATH] |
Step I : Recovering an [MATH] -definable map [MATH] from [MATH] The results on strongly large sets from Section 2.3 are used here. Note that in the next lemma, if the set [MATH] were [MATH] -definable, we would already have proved Theorem 5.1 (with [MATH] [MATH] , and [MATH] ). |
Lemma 5.2 There are a closed [MATH] -definable set [MATH] , with [MATH] , a definable set [MATH] , with [MATH] , and an [MATH] -definable map [MATH] , such that |
[EQUATION] Moreover, [EQUATION] for some [MATH] , for [MATH] , with [MATH] and [MATH] Proof. By Corollary 2.7 [MATH] is the union of a full set [MATH] and a set of dimension [MATH] . Let [MATH] . Since [MATH] is full, |
[MATH] and hence [MATH] . Since also [MATH] , we obtain [MATH] . Now, by Lemma 2.5 , there is a definable set [MATH] of dimension [MATH] and an [MATH] -definable map [MATH] that agrees with [MATH] on [MATH] Let |
[EQUATION] and for [MATH] [EQUATION] Let also [MATH] By (D3) [MATH] , and since [MATH] [MATH] . Similarly, [MATH] , for [MATH] It follows from (D3) that [MATH] Finally, since [MATH] , we have [MATH] |
For the rest of this section, we fix [MATH] and [MATH] as above, and use their properties without any specific mentioning. Note that since [MATH] , it follows that [MATH] [MATH] , and, for [MATH] [MATH] |
Step II: Group-like properties of [MATH] Here we prove the existence of an [MATH] -definable set [MATH] , with [MATH] , on which [MATH] is continuous and behaves like a group operation, with [MATH] . This is done through a series of lemmas. |
Lemma 5.3 There is an [MATH] -definable set [MATH] , with [MATH] , such that [MATH] is continuous and [MATH] Proof. By o-minimality, there is an [MATH] -definable set [MATH] , which is a finite union of [MATH] -cells, with [MATH] , such that [MATH] is continuous. We claim that [MATH] , for some set [MATH] which is dens... |
[EQUATION] Since [MATH] , we have that [MATH] is injective. Since also [MATH] , it follows from (D4) that [MATH] . Hence [MATH] . Moreover, [MATH] Let |
[EQUATION] By (D3) [MATH] , and hence [MATH] . By Fact 2.1 , since [MATH] is a finite union of [MATH] -cells, [MATH] is dense in [MATH] . Moreover, [MATH] , as required. Now, since [MATH] [MATH] is closed and [MATH] is continuous, it follows that [MATH] |
Lemma 5.4 There is an [MATH] -definable set [MATH] with [MATH] , such that [MATH] is injective in each coordinate. Proof. It suffices to find an [MATH] -definable set [MATH] with [MATH] , such that [MATH] is injective in the second coordinate. One can then similarly find [MATH] with [MATH] and [MATH] injective in the f... |
Suppose towards a contradiction that there is no such [MATH] . For every [MATH] , let [EQUATION] Then the set [EQUATION] is [MATH] -definable. So for any [MATH] [MATH] is not injective on any [MATH] -definable subset of [MATH] of co-dimension [MATH] . So, by assumption and (D3) [MATH] . Since [MATH] is dense in [MATH] ... |
Lemma 5.5 There is an [MATH] -definable set [MATH] with [MATH] , such that for every [MATH] [EQUATION] Proof. By o-minimality, there is an [MATH] -definable set [MATH] , which is a finite union of [MATH] -cells, with [MATH] , such that both maps [MATH] and [MATH] are continuous on [MATH] It is thus enough to prove that... |
We observe that for every [MATH] with [MATH] [MATH] and [MATH] , equation (*) holds, since both of its sides equal [MATH] . Hence, if, for every [MATH] , we let |
[EQUATION] and [EQUATION] then (*) holds on the set [EQUATION] Since [MATH] [MATH] and, for [MATH] [MATH] , it follows easily from (D4) that for every [MATH] and [MATH] [MATH] and [MATH] Since also [MATH] (D3) implies that [MATH] , and hence [MATH] . Thus [MATH] , and by Fact 2.1 [MATH] is dense in [MATH] |
We can refine the set [MATH] in order to achieve two additional properties. Corollary 5.6 Let [MATH] be as in Lemma 5.5 . Then, there is an [MATH] -definable set [MATH] , such that |
[MATH] [MATH] is continuous, and injective in each coordinate, [MATH] for every [MATH] [MATH] for every [MATH] [MATH] Proof. Let [MATH] be as in Lemma 5.3 . Define |
[EQUATION] and [EQUATION] Since [MATH] , we obtain [MATH] and [MATH] . Define also [EQUATION] Since [MATH] , we obtain [MATH] . The desired set [MATH] is the intersection of [MATH] [MATH] and the set obtained in Lemma 5.4 |
For the rest of this section, we fix the sets [MATH] and [MATH] as above, and use their properties without any specific mentioning. |
Step III: Extracting an [MATH] -definable set [MATH] using [MATH] In this step, we use [MATH] to recover a suitable [MATH] -definable set [MATH] with [MATH] . The work from Section plays an essential role here. The suitability of [MATH] will be evident in Step IV. |
Lemma 5.7 The set [MATH] is [MATH] -definable with [MATH] Proof. By Corollary 3.8 [MATH] is L-definable, so we need to show that [MATH] . By cell decomposition, [MATH] is a finite union of cells. Let [MATH] be the union of all [MATH] -cells in this decomposition. We write [MATH] for [MATH] . Since [MATH] is [MATH] -def... |
[EQUATION] and hence [MATH] . Since [MATH] , we obtain [MATH] . By Fact 2.1 [MATH] is dense in [MATH] . On the other hand, [EQUATION] |
That is, [MATH] is dense in [MATH] , and hence so is [MATH] Step IV: Constructing a definable embedding [MATH] In this step, we embed [MATH] into [MATH] , after proving the key property (*) from the introduction. For every [MATH] and [MATH] , the set |
[EQUATION] is definable. Lemma 5.8 For every [MATH] , there is unique [MATH] , such that [MATH] Proof. Let [MATH] , where [MATH] . Recall that [MATH] . Let |
[EQUATION] Then [MATH] . Moreover, for every [MATH] , we have [EQUATION] That is, for [MATH] , we obtain [MATH] . It follows that [MATH] |
The uniqueness of [MATH] is clear, since otherwise we would obtain two sets [MATH] and [MATH] both contained in [MATH] and having co-dimension [MATH] in [MATH] , a contradiction. |
We now consider the map [MATH] given by [EQUATION] Recall that for every [MATH] [MATH] Claim 5.9 [MATH] is injective. Proof. Suppose that for [MATH] , we have [MATH] . Then [MATH] and [MATH] . Therefore, |
[EQUATION] For [MATH] in that intersection, we have [MATH] , and by injectivity of [MATH] in the first coordinate, [MATH] Claim 5.10 |
For every [MATH] [MATH] . In particular, [MATH] Proof. Let [MATH] . Then for every [MATH] [MATH] . So, [MATH] . Since [MATH] , the result follows. |
Since [MATH] , we have [MATH] . Since also [MATH] , we obtain [MATH] . Therefore, by Claim 5.10 [MATH] Step V: Concluding the proof of Theorem 5.1 |
It remains to show the following statement. Lemma 5.11 There is an [MATH] -definable set [MATH] with [MATH] and [MATH] , such that for every [MATH] |
[EQUATION] Proof. We let [EQUATION] Clearly, [MATH] . We prove [MATH] . Recall that [MATH] , and hence [MATH] . So, it suffices to prove that [MATH] . Let [MATH] . Since [MATH] , we have [MATH] . Let [MATH] . Then for every [MATH] [MATH] , and |
[EQUATION] Hence, the set [EQUATION] belongs to [MATH] and has co-dimension [MATH] in [MATH] , as required. Now let [MATH] . For [MATH] , denote [MATH] . So [MATH] . By injectivity of [MATH] in the second coordinate, [MATH] has co-dimension [MATH] in [MATH] Hence the set |
[EQUATION] is non-empty. Take any [MATH] . Then [EQUATION] and hence [MATH] , as required. This ends the proof of Theorem 5.1 , and hence, by Theorem 4.4 , also that of Theorem 1.1 |
6. Expansions by dense independent sets In this section, we let [MATH] be an o-minimal expansion of an ordered group, [MATH] a dense [MATH] -independent set, and [MATH] . We let [MATH] be the large dimension coming from |
, as described in Section 2.4 . Note that the assumption that [MATH] expands a group is not due to any reasons pertaining the current work, but only because the accounts |
and that analyze this pair work under it. Theorem 1.2 will follow from Theorem 1.1 and the following theorem. Theorem 6.1 Every definable group is definably isomorphic to a strongly large group. |
The rest of this section is devoted to proving Theorem 6.1 . The proof is based on the cone decomposition theorem from . The terminology of Section 2.4 applies here. A simplified formulation of the cone decomposition theorem is that every definable set [MATH] is a small union of sets of the form [MATH] , where [MATH] i... |
Definition 6.2 (Cones) A set [MATH] is a [MATH] -cone [MATH] , if there is a definable set [MATH] , where [MATH] and every [MATH] is a supercone, and an [MATH] -definable continuous map [MATH] , where [MATH] is cell, such that |
(1) for every [MATH] [MATH] (2) [MATH] (3) [MATH] is injective. cone is a [MATH] -cone for some [MATH] Remark 6.3 It is important to note that if [MATH] is a cone as above, then for every [MATH] , the set [MATH] is a full set (as in Section 2.3 ). Indeed, the map [MATH] is [MATH] -definable continuous and injective. By... |
Fact 6.4 (Cone decomposition theorem) Every definable set is a finite disjoint union of cones. Proof. This is a consequence of the cone decomposition theorem in |
and subsequent work in . A detailed proof is given in 16 , Fact 4.7] . In that reference the universe of [MATH] is assumed to be [MATH] , but this played no role in the particular proof. |
We will need a further decomposition as follows. Claim 6.5 Every [MATH] -cone is a finite disjoint union of [MATH] -cones [MATH] , with [MATH] , such that: |
(1) every [MATH] has all its coordinates distinct, (2) [MATH] is either finite, or every coordinate projection of [MATH] is infinite. |
Proof. We first show that every [MATH] -cone [MATH] can be written as a finite disjoint union of [MATH] -cones satisfying (1). Let [MATH] , with [MATH] and [MATH] , as in Definition 6.2 . We work by induction on [MATH] . For [MATH] , the result obviously holds. Let [MATH] , and consider the set [MATH] of all those elem... |
[EQUATION] and [EQUATION] and define [MATH] , where [MATH] , and [MATH] with [MATH] . Then [MATH] , with [MATH] . By inductive hypothesis, the result follows. |
Now, we show that every [MATH] -cone that satisfies (1) can be written as a finite disjoint union of sets satisfying (1) and (2). Let [MATH] be as above. We work again by induction on [MATH] . If [MATH] , the result obviously holds. Let [MATH] , and suppose that some coordinate projection of [MATH] is not infinite, say... |
with [MATH] . Then each [MATH] is still a [MATH] -cone satisfying (1), and [MATH] . By inductive hypothesis, the result follows. |
We will also need the following lemma. Lemma 6.6 Every [MATH] -cone can be definably embedded into [MATH] Proof. Let [MATH] be a [MATH] -cone, with [MATH] and [MATH] as in Definition 6.2 . We first embed [MATH] into [MATH] . By Fact 2.11 , there are [MATH] which are [MATH] -independent over [MATH] . Define [MATH] via |
[EQUATION] By choice of [MATH] , it follows that [MATH] is injective. Now, since [MATH] is injective, we can embed [MATH] into [MATH] via [MATH] |
Corollary 6.7 Let [MATH] be a definable set of dimension [MATH] . Then there is a definable bijection [MATH] with [MATH] Proof. By cone decomposition, [MATH] is a finite union of cones. By Lemma 6.6 , each of the cones can be definably embedded into [MATH] . Then [MATH] can be definably embedded into finitely many disj... |
Corollary 6.8 Let [MATH] be the union of a full set and a set of smaller dimension. Then [MATH] is in definable bijection with a strongly large set. |
Proof. Suppose [MATH] and [MATH] , with [MATH] full and [MATH] . Since [MATH] , and using Corollary 6.7 for [MATH] , we can easily embed [MATH] into some [MATH] via an [MATH] , such that [MATH] . Therefore [MATH] is strongly large. |
Note that we only used that [MATH] is strongly large in the above proof. Proof of Theorem 6.1 Let [MATH] be a definable group, with [MATH] and [MATH] . For [MATH] , we write [MATH] for [MATH] . By Fact 6.4 and Claim 6.5 [MATH] is a finite union of cones [MATH] , each of the form [MATH] , where [MATH] satisfies Claim 6.... |
Now fix a [MATH] -cone [MATH] among [MATH] , with [MATH] infinite and [MATH] maximal such. By Claim 6.5 (1) & (2), we can find two distinct elements [MATH] with all their [MATH] coordinates distinct. Let [MATH] be the set of all those [MATH] coordinates. Let also [MATH] be a finite parameter set that is used to define ... |
Case: [MATH] Since all of [MATH] are [MATH] -cones, we may write [MATH] , where [MATH] , for some [MATH] . Let [MATH] and [MATH] be so that |
[EQUATION] Since [MATH] , there must be [MATH] . Say [MATH] (if [MATH] , the argument is symmetric). By injectivity of [MATH] [MATH] , contradicting the fact that [MATH] is [MATH] -independent over [MATH] |
Case: [MATH] We need the following claim. Claim. There are a [MATH] -cone [MATH] among the [MATH] ’s, with [MATH] , a tuple [MATH] , and a triple [MATH] , such that [MATH] is [MATH] -independent, and |
[EQUATION] Proof of the claim. Let [MATH] . Then [MATH] is a supercone in [MATH] . By (D6) , there is a definable set [MATH] with [MATH] and an [MATH] -definable map [MATH] such that the map |
[EQUATION] agrees with [MATH] on [MATH] . By o-minimality, there is an open cell [MATH] , such that [MATH] is continuous. By 17 , Lemma 4.16] [MATH] is a supercone in [MATH] , and hence the set [MATH] also has dimension [MATH] . By Fact 2.11 , there is [MATH] , which is [MATH] -independent over [MATH] . Moreover, by 17... |
[EQUATION] Since [MATH] , it follows that [MATH] is also [MATH] -independent over [MATH] and has dimension at least [MATH] . Hence [MATH] . ∎ |
Let [MATH] and [MATH] be as in the claim. By maximality of [MATH] , it must be that [MATH] , for some [MATH] . Since all [MATH] coordinates of [MATH] are distinct, there must be [MATH] . Say [MATH] By (*), |
[EQUATION] and hence [EQUATION] contradicting the fact that [MATH] is [MATH] -independent over [MATH] . ∎ 7. A future direction There are many tame expansions of o-minimal structures that support a nice notion of dimension, and hence where at least the methods of this paper could apply. Examples include real closed val... |
or . Assume that [MATH] is a distal structure, and let [MATH] be an expansion of [MATH] , which is NIP, but not distal. This is the case, for example, with all pairs [MATH] mentioned in Section 2.4 (see |
). Define the distal closure operator [MATH] as follows: [EQUATION] Work from implies that for a dense pair of real closed field, a type [MATH] is small (that is, it contains a small formula) if and only if [MATH] is distal. Combined with work from |
, we obtain that [MATH] in this setting is a pregeometry, and that the corresponding [MATH] -dimension coincides with the large dimension (as in Section 2.4 ). The proposed direction is to explore further expansions N where [MATH] is a pregeometry, and, if [MATH] is o-minimal, to check whether axioms (D1)-(D6) for the ... |
# Source: arxiv 1805.11532 # Title: The GraftalLace Cellular Automaton # Sections: all # Downloaded: 2026-03-02T08:58:03.033778+00:00 |
The GraftalLace Cellular Automaton Abstract. We introduce our GraftalLace Cellular Automaton in short GLCA which is a new one-dimensional cellular automaton on the regular square lattice. It makes a monochromatic infinite directed graph which evolve deterministically row by row, by a defined rule and a single initial r... |
Definition of GLCA I invented GLCA in 1991 inspired by the articles of Scientific American magazine about elementary cellular automata of Stephen Wolfram [W84,W02] and graftal trees otherwise recursive fractal plants of Aristid Lindenmayer [D86,PL90]. Graftal is a combination of two words: graph + fractal. |
GLCA connects root patterns with branch patterns through a junction (otherwise a grid point of the square lattice) by a defined rule. Both patterns are triplets of arcs formed by the incoming and the outgoing arcs of a junction from and into the same horizontal position with the left and right neighbour cells. Usually ... |
1.1 Formal definition of GLCA GLCA operating with a chain of deterministic finite automata (DFA) and can be represented as a 4-tuple [MATH] , where [MATH] is an octal alphabet (cell states), [MATH] is the local transition function, [MATH] is a function to define the cell neighbourhood with bit operations and [MATH] is ... |
In the next subsections we show two bit operating functions: [MATH] and [MATH] to define the neighbourhood of the cells because transition function in GLCA unlike other cellular automata only partly influences the states of the next cells (3 cells, one at the same horizontal position with the left and right neighbour c... |
The array increases maximum 2 cells in each time steps [MATH] . Evolution of GLCA is represented by a sequence of finite configurations [MATH] given by the global mapping, [MATH] |
[EQUATION] Figure 2. A rule as an octal number [MATH] means how to connect all the root patterns (upper triplets) with branch patterns (lower triplets |
in reverse order binary code) through the junction (centre point). 1.2 Basic definitions Our cell space is the regular simple upright square lattice. All grid points are cells called the junctions . We use the Cartesian coordinate system with upside-down y-coordinates. We only allow connections between a cell and its c... |
Both patterns form binary triplets which have 8 possible variants denoted by an octal digit therefore we use the octal alphabet [MATH] . The state of a cell ( [MATH] ) specified by its incoming arc triplet, the root pattern. By choosing an eight-digits long octal number ( [MATH] ) we get a rule for our GLCA which tells... |
For practical reasons we denote the branch patterns with a reverse order binary number because in the next time step (next row of the evolving pattern) branch arcs become root arcs in another arrangement where every arc belongs to different junctions in a reverse order. Both patterns overlap each other. Figure 3 shows ... |
[EQUATION] Figure 3. Overlapping roots and branches. We show 3 cells in the middle row (X,Y,Z) with all of their possible connections. |
Their branches make a new root pattern by the incoming arcs of cell S. [EQUATION] Figure 4. Data representation of overlapping branch patterns |
from 3 junctions (X,Y,Z above) into 5 next junctions (below). Binary digits in a new combination represent a new root pattern: [MATH] |
The rule defines the corresponding branch pattern for any potential root patterns otherwise for any potential states of the cells: [MATH] which means 3 possible outgoing connecting arcs from the junction. These arcs form an octal digit as a reverse order binary triplet by the highest bit: [MATH] , the middle bit: [MATH... |
One branch pattern (outgoing triplet of arcs from a junction) partly influences the states of its 3 different neighbour cells in the next row by changing their corresponding bits. The states of the cells come from their root pattern as their incoming connecting arcs from 3 different cells into a junction by the highest... |
Our rule is assigning all possible [MATH] values into not necessarily different [MATH] values. The total number of the possible rules are [MATH] [MATH] . We avoid growing branches from nothing therefore the last digit of the rule is always equal to zero. The number of the remaining rules is [MATH] [MATH] |
1.3 Bit operating functions In this section we show how the local transition function creates the states of the new cells in the next time step automatically (1). We define a new bit operating function [MATH] which gives back the value of the [MATH] component of an octal digit [MATH] |
For example: [MATH] [EQUATION] Let’s consider another bit operating function [MATH] which means let the [MATH] th bit of the octal number [MATH] is equal to bit [MATH] . For example: if [MATH] then [MATH] means [MATH] , and after that [MATH] means [MATH] . Now we can show how the local transition function creates the b... |
The following branches partly influence the states of 3 different cells in the next row: [EQUATION] See Figure 4 for data representation of a new root pattern (new state of the cell S below cell Y) made by combined bits of different branch patterns. Branch patterns with reverse order bits: [MATH] [MATH] [MATH] automati... |
Symmetric fractal patterns We can find all the pattern groups of Wolfram’s classification (Class I-IV) among GLCA patterns otherwise the evolution of the patterns leads to homogenous, regular, chaotic and complex patterns. See Figure 5 and |
We show how can we realize Pascal triangle modulo [MATH] symmetric fractal pattern by the monochromatic GLCA. See Figure 7 The Sierpiński triangle (Pascal triangle modulo 2) pattern can be realized in many ways. For example by applying an XOR binary operation or an iterated function system (IFS) rule onto a binary squa... |
GLCA also gives other possibilities to realize this fractal pattern. The simplest one, Rule [MATH] can be drawn from any single root arc. The rule means draw two vertical branch arcs from single arcs and do not draw in other cases. See Figure 8 . Rule [MATH] [MATH] and [MATH] also make this fractal in another way. |
We can realize Pascal triangle modulo 3 pattern in many different ways also. As an IFS fractal [W02], by recursive curves otherwise by Hamiltonian paths or Hamiltonian cycles [K17a,K17b]. With Wolfram’s automaton we have to use more colours [W84,W02] (3 colours, totalistic rule, code 420) unlike my monochromatic GLCA p... |
Searching for reversible rules A reversible cellular automaton is a system that is deterministic in both directions in terms of time. It is also called invertible cellular automaton. In GLCA it means if we change the direction of all arcs of the mini trees into reverse we can continue the drawing at the other side of a... |
In reversible rules we have to avoid growing branches from nothing therefore the last digit of the rule is always equal to zero. We have to use assignments amongst root patterns and branch patterns with one-to-one correspondence. We have 7 different patterns so the maximum number of these unambigous assignments are equ... |
By leaving odd numbers of digits at their place-values in the octal rule number and changing the remaining digits pairwise by mutuality of the number of the place value and the correlating digit we get the following sum of binomials: [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] . In this case the rule number does not depe... |
In the remaining cases we get a different rule number by changing the direction of the drawing. We get the correlating rule pair by replacing the digit values with the place-values of the rule number for example: Rule [MATH] and Rule [MATH] are correlating pairs. |
Extensions and variations of the basic idea By using the same 3 arcs long root and branch patterns and bichromatic arcs (2 drawing colours and 1 background colour) the triplets can be described as 3-digit long numbers in ternary numeral system. In this case we combine [MATH] root patterns with also [MATH] branch patter... |
We recommend Wolfram’s method the totalistic rules to define the assignments in an easier way. Instead of defining branch patterns for every possible root pattern it is enough to assign branch patterns to groups of root patterns as hues or densities of arcs. These hues or densities are equal to the sum of the digits of... |
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