On the Existence of One-Point Time on an Oriented Set
GRUSHKA YA. I.
Institute of Mathematics NAS of Ukraine,
3, Tereschenkivska st. Kyiv-4. 01024,
UKRAINE
Abstract: - The oriented set notion is the elementary fundamental concept of the theory of changeable sets. In turn,
the changeable set theory is closely related to Hilbert's sixth problem. From the formal point of view, any oriented set
is a simple relational system with a single reflexive binary relation. Such mathematical structure is the simplest
construction, within the framework of which it is possible to give a mathematically strict definition of the time
concept. In this regard, the problem of the existence of time with given properties on an oriented set is very
interesting. In the present paper, we establish the necessary and sufficient condition for the existence of one-point
time on an oriented set. From the intuitive point of view, any one-point time is the time related to the evolution of a
system, which consists of a single object (for example, from a single material point). The main result of the paper
provides that the one-point time exists on the oriented set if and only if this oriented set is a quasi-chain. Also, using
the obtained result, we solve the problem of describing all possible images of linearly ordered sets, which naturally
arises in the theory of ordered sets.
Key-Words: - Binary relations, reflexive relations, oriented sets, changeable sets, time, ordered sets, quasi-ordered
sets.
Received: April 14, 2023. Revised: November 29, 2023. Accepted: December 15, 2023. Published: December 31, 2023.
1 Introductory Remarks
The subject of this article is closely related to the
theory of changeable sets. In turn, this theory is
connected with the famous sixth Hilbert problem, that
is the problem of mathematically strict formulation for
the fundamentals of theoretical physics. The last
problem was posed in 1900, but it remains very
relevant today, [1], [2], [3], [4], [5], [6], [7]. From the
intuitive point of view, changeable sets can be
interpreted as sets of objects, which can be in the
process of permanent transformations. Namely, these
objects can change their properties, appear or
disappear, break down into several parts or,
conversely, unite into a single unit. Moreover, the
evolution picture of a changeable set may depend of
the area of observation or reference frame. The
problem of the creation the mathematical theory of
changeable sets (that is the “sets” possessing the
properties listed above) emerged in particular in the
papers [8], [9], [10], [11], [12], [13]. In the papers of
the author of this article the theory of changeable sets
was developed on the mathematically strict level. The
most complete and systematic presentation of this
theory can be found in the preprint, [14]. For more
information about scientific papers in peer-reviewed
journals, where the foundations of the changeable set
theory were first published, see the reference list in the
end of preprint.
The notion of oriented set is the basic most
elementary concept of the theory of changeable sets.
Oriented sets were introduced in [15], as the most
simple abstract models of the collections of evolving
objects in the framework of one (fixed) reference
frame ([14], Section 1). Moreover, in the
aforementioned papers it was introduced the concept
of time on oriented sets. As well in the article [15], (in
Theorem 4.1) the sufficient condition of existence of
one-point time for oriented sets is established ([14],
Theorem 1.3.1). Note that from the intuitive point of
view, one-point time should be understood as the time
associated with the evolution of a system consisting of
only one object (for example, from one material
point). Emphasize that Theorem 4.1 from [15], gives
only sufficient conditions for the existence of
one-point time. That is why in the paper [14],
(Problem 1.3.1) the problem of detection of necessary
and sufficient conditions for existence of one-point
time on an oriented set is posed. Below in this paper,
the solution of the above problem is presented.
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Namely, we specify the properties for the oriented set
to be able to define the one-point time on it. Using the
obtained result, we solve the problem of describing all
possible images of linearly ordered sets. Such a
problem naturally arises in the theory of ordered sets.
2 On Oriented Sets and
One-point Time
Definition 1. Let, be any nonempty set
( ).
An arbitrary reflexive binary relation on
(that is a relation satisfying ) we
name an orientation, and the pair we
call an oriented set. In this case the set is named
the basic set or the set of all elementary states of the
oriented set and it is denoted by . The
relation we name the directing relation of changes
(transformations) of , and denote it by
.
In the case where the oriented set is known in
advance we use the notation instead of
. For the
elements the record should be
understood as “the elementary state is the result of
transformations (or the transformation offspring) of
the elementary state ”.
Let be an oriented set.
Definition 2. The nonempty subset
is referred to as transitive in if for any
such, that and we have
. The transitive subset is referred
to as chain in if for any at least one of
the relations or is true.
Oriented set is called a chain oriented
set if the set is the chain of , that is if the
relation if transitive on and for any
at least one of the conditions
or is satisfied (note that this is the case, where
the oriented set is a linearly quasi-ordered set).
Recall that a (partially) ordered set is an ordered pair
of kind with reflexive, asymmetric and
transitive binary relation on . The pair is
called an linearly ordered set if the following
additional condition holds:
(LnO) for every it is performed at least one
of the correlations or .
Definition 3. Let be an oriented set and
be a linearly ordered set. A mapping
is referred to as time on if the
following conditions are satisfied:
1. For any elementary state
there exists an element such that .
2. If , and
, then there exist elements such that
, and (this means
that there is a temporal separateness of successive
unequal elementary states).
In this case:
• The elements we call the moments of
time
• The pair we
name by chronologization of .
We say that an oriented set can be
chronologized if there exists at least one
chronologization of . It turns out that any oriented
set can be chronologized. To make sure this we may
consider any linearly ordered set , which
contains at least two elements and put:
It is easy to verify that the conditions of Definition 3
for this function are satisfied. More nontrivial
methods to chronologize an oriented set were
considered, in particular, in [15].
Definition 4. Let be an oriented set and
be a linearly ordered set.
1. The time is called quasi
one-point if for every the set is a
singleton.
2. The time is called one-point if the
following conditions are satisfied:
(a) The time is quasi one-point;
(b) for any the conditions
, and , lead to
. We say that an oriented set can be
chronologized quasi one-point / one-point if there
exists at least one chronologization
of with quasi one-point /one-point time
(correspondingly). In this case we name the
chronologization as quasi one-point /one-point
(correspondingly).
Example 1. Let us consider an arbitrary
mapping ( ), where is some
connected subset of Real axis . This mapping can be
interpreted as equation of motion of a single material
point in the space . This mapping generates the
oriented set
, where
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and for
, the correlation
is valid if and
only if there exist such, that ,
and . Consider the following
set-valued mapping:
It is easy to verify, that the mapping satisfies
the conditions of Definition 3 and Definition 4
(item 2). Consequently is an one-point time on
.
Example 1 makes clear the definition of
one-point time. It is evident, that any one-point time is
quasi one-point. Examples contained in the paper
[15], show that the inverse statement is not true in the
general case ( [14], Example 1.3.2).
Theorem 1. (ZF+LO, [15]). Any oriented set
can be quasi one-point chronologized.
Note that proof of Theorem 1 can be found
also in [14] (see Theorem 1.3.2).
Remark 1. Proof of Theorem 1 uses the
Linear Ordering principle (LO) in addition to
Zermelo–Fraenkel axiomatic system (ZF). This
principle asserts that any set can be linearly ordered. It
is evident that the above principle follows from the
famous well-ordering Zermelo’s theorem, and
therefore, from the axiom of choice (AC). But it is
known that LO-principle also follows from Ultrafilter
theorem of Tarski (UFT) and, moreover, it is logically
weaker than this theorem and therefore than the axiom
of choice, [16]. On the relationship between LO and
AC see, also, [17].
Theorem 2 (ZF+LO, [15]). Any chain
oriented set can be one-point chronologized.
Note that the proof of Theorem 2 can be also
found in [14]. It turns out that Theorem 2 is not
reversible. And the next example demonstrates the
existence of non-chain oriented sets, which can be
one-point chronologized.
Example 2. Consider the function
, defined by the formula:
cos sin .
According to Example 1, using this function, we may
construct the oriented set . This oriented set can
be one-point chronologized by means of the time:
.
At the same time, this oriented set is not a chain,
because the binary relation
is not transitive on
. Indeed, consider the points:
,
,
. For these points we
have: and
,
but the correlation
is false.
The above facts generate the following
problem:
Problem 1. Find necessary and sufficient
conditions of existence of one-point chronologization
for oriented set.
Note that Problem 1 was also posed in [14],
(Problem 1.3.1). The main aim of the present paper is
to give the solution of Problem 1.
3 Quasi-chain Oriented Sets and
Formulation of Main Theorem
Notation 1. On any oriented set we introduce the
following additional binary relation:
For every we note
if and only if:
and
.
In the cases where it does not lead to
misunderstanding we use the notation
instead
of the record
.
Notation 2. Let be an arbitrary set and
( ) be any binary
relations on . Further for we use
the abbreviated notation:
for indication the fact that:
.
Assertion 1. Let be an oriented set,
be a linearly ordered set and
be an one-point time on . Then for any
the conditions:
and
lead to the inequality:
Proof. Indeed, suppose that is an oriented
set, is a linearly ordered set and
is an one-point time on . Let the elements
be such that ,
and
. Assume the contrary:
. Then, according to Definition 4 (item 2), from the
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conditions , and it
follows that . But the last correlation is in
contradiction to the condition
. Hence the
assumption about is false. Therefore we have
.
Definition 5. The oriented set is called
quasi-chain if and only if the following conditions are
satisfied:
(QL1) For any it holds at
least one from the correlations or .
(QL2) For every the
condition
ensures the
correlation
(quasi-transitivity).
Remark 2. It is easy to prove that the
transitivity of the binary relation on the oriented
set implies its quasi-transitivity. It turns out that
the inverse statement in general is not valid. Example
2 shows that there exist the oriented set
such that the relation
is quasi-transitive but not
transitive. So quasi-chain oriented set must not be
chain.
The main result of this paper is the following
theorem.
Theorem 3 (ZF+UFT). An oriented set
can be one-point chronologized if and only if it is a
quasi-chain.
Remark 3. We emphasize that proof of the
necessity for Theorem 3 does not require the
Ultrafilter Tarski theorem (UFT). This theorem is
needed only for the proof of sufficiency of the
condition, pointed out in Theorem 3.
The proof of Theorem 3 is divided into two
main lemmas. Lemma 1 in the next section assures the
necessity for Theorem 3, whereas Lemma 2 (see
below) provides the sufficiency.
4 Proof of Necessity for
Theorem 3
Lemma 1. If the oriented set can be one-point
chronologized then it is a quasi-chain.
Proof. Let be a linearly ordered
set and be an one-point time on the
oriented set .
1. First we will validate the condition
(QL1). Chose any . By Definition 3
the time points must exist such, that
, . Since is a linearly
ordered set then for at least one of the
inequalities must be fulfilled or . In
Accordance with Definition 4, in the case we
obtain . Similarly in the case we
deduce .
2. Now we validate the condition
(QL2). Consider any elements
such, that
. Consider
any such, that ,
(by Definition 3 such exist). Since ,
then, according to Definition 3 the time points
must exist such that , and
. Taking into account the correlations
, and
, as well as Assertion
1, we obtain, . Similarly from the correlations
, and
we deduce
. Therefore the following inequalities are
performed:
That is why . Thus we have:
(1)
In accordance with the statement, proven in the
item 1, at least one from the correlations or
must hold. Assume, that . Then, by
Definition 3 the elements must exist such
that , and . But the
last inequality is in a contradiction to (1). Hence, the
correlation is impossible. Thus the only
possible one it remains the correlation
, that it
was necessary to prove.
The proof of the sufficiency for Theorem 3 is
much more complicated. First of all we need to work
out some auxiliary technical results for this purpose.
This work will be done in the next section.
5 Some Auxiliary Technical
Results
5.1 Some Additional Properties of
Quasi-chain Oriented Sets
Assertion 2. Let, be a quasi-chain oriented set
and be arbitrary elementary
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states of . Then the following properties are
performed:
(QL3) If
then .
(QL4) If
then .
(QL5) If
then .
(QL6) If
then
.
Proof. The proofs of the properties
(QL3)–(QL6) are listed below.
(QL3). Let and
. Assume that the correlation
is false (IE ). Then, taking into
account the fact that the oriented set is
quasi-chain, we get
. Thus, we have,
. Hence, by Definition 5
(condition (QL2)) we get,
, which is in a
contradiction to the correlation . Therefore
assumption about is false. So we have
.
(QL4). Suppose that
and
. Then, by Definition 1,
we have,
. Thence, using
Property (QL3), we obtain .
(QL5). If we assume that
,
then we will have
. Thence,
applying Property (QL3), we obtain .
(QL6). If we suppose that
, then we will deliver
. Thence, by Definition 5 (condition (QL2)),
we deduce
.
5.2 Finite-repeating Time on Oriented
Sets
Definition 6. Let be a linearly ordered
set and be an oriented set.
• The time will be named as
finite-repeating if and only if for every
the following condition is fulfilled:
(where is the cardinality of a set ).
Moreover, the number:
Rp
will be refereed to as repeatability of the time
relatively the element .
• Let . The time is named as
-repeating if and only if the time is
finite-repeating and
Rp
Notation 3. Let be a
finite-repeating time on the oriented set . For
every we note:
max
min
where maximum and minimum should be understood
it the sense of the linearly ordered set .
Assertion 3. Let be a linearly
ordered set and be a finite-repeating
one-point time on the oriented set . Then for any
the following properties are
holding:
(FR1)
. If, in addition,
Rp then
.
(FR2) The correlation is true if and
only if
. If, in addition,
then if and only if
.
(FR3)
if and only if
.
(FR4) If, in addition, the time is
-repeating with then if and only if
.
Proof. (FR1): Let . Then
according to Notation 3, we have
min max
. If, in addition, Rp then
the set contains at least two
elements. So minimum of this set is less then
maximum.
(FR2): First we suppose that
and .
Then in the case we have the
inequality
according to Property
(FR1). Hence we will consider that . Since
and , then, by Definition 3, the time
points exist such that ,
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and . Therefore:
min
max
So, for every it is performed the
following implication:
(2)
Thus, in the both cases for any we
have the implication:
(3)
Conversely, suppose that
. Put:
Then in accordance with Notation 3, we have,
, and . Hence, by
Definition 4, we deduce . Thus for every
we have the implication:
(4)
The implications (3) and (4) assure the desired
equivalence:
.
If we assume that, in addition, then
from (2) and (4) we deliver the equivalence:
.
(FR3): Let
. Assume that
. Then according to Property (FR2), we obtain
the correlation , which contradicts to
. Therefore,
.
Conversely, suppose that
.
Then, applying Property (FR1), we deliver
. Hence, according to
Property (FR2), we obtain . Assume that the
condition also is performed. Then by
Property (FR2), we get the inequality
, which contradicts to the inequality
. That is the assumption about
is wrong. That is why we have
.
(FR4): In the case Property (FR4)
follows from Property (FR2). In the case this
property follows from Property (FR1).
Remarks on the idea of proof the sufficiency of
Theorem 3. It turns out that it is technically easier to
prove the existence of -repeating one-point time on
the quasi-chain oriented set . Taking into account
this situation, we can take the set
as the set of time points and consider the
mapping:
(5)
Then for the proof of desired result it is sufficient to
find the linear order relation on , which turns the
mapping (5) into a one-point time. Further we will
consider that the desired order satisfies the
following natural additional condition:
(6)
Assume that the mapping (5) is an one-point time.
Taking into account convention (6), for every
we obtain the equalities:
(7)
From the equalities (7) and properties (FR2),
(FR3) (see. Assertion 3) it follows that the desired
order on must have the
following properties:
• If and then
.
• If and
then
.
6 Proof of Sufficiency for
Theorem 3
For proving the main result of this section we need the
following auxiliary assertion:
Assertion 4 ([16]). Let be a partially
ordered set. Then the linear order on the set
exists such that
.
Emphasize that the inclusion of binary relations in
Assertion 4 should be understood in set-theoretic
sense, that is the record “ ” means that for
any the correlation leads to
.
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It is known that Assertion 4 is a consequence of
Ultrafilter Tarski theorem (UFT). In turn UFT follows
from the axiom of choice (AC), moreover UFT is
logically weaker than AC. So Assertion 4 also is
logically weaker than AC. But, from the other hand, it
is known that this Assertion can not be obtained from
Zermelo–Fraenkel axiomatic system without AC (for
details see [16], Theorem 2.18 and Proposition 4.39).
The next lemma ensures the sufficiency for
Theorem 3.
Lemma 2. If the oriented set is a quasi-chain
then it can be one-point chronologized. Moreover
there exists the chronologization of
with -repeating one-point time .
Proof. Let be quasi-chain oriented set.
Denote:
First we introduce the binary relation on the set
by the following rule:
For any , we
write if and only if at least one of the
following conditions is satisfied:
(Preo1) (that is and );
(Preo2) , , ;
(Preo3)
, , .
Also, we note if and only if and
.
The introduced relation is obviously reflexive
(ie ). Moreover, we are going to prove
that this relation has the following property of the
“weak” transitivity:
(WT) If and
then .
Indeed, let and
, where , ,
. Since ,
Condition (Preo1) can not be performed for the
elements and . Hence one of the conditions
(Preo2), (Preo3) must be fulfilled. If Condition
(Preo2) is fulfilled, we have, , ,
. Next, since , Condition (Preo1) can not be
performed for the elements and . Condition
(Preo2) also can not be performed for the elements
and , because . Therefore Condition
(Preo3) is fulfilled, that is
, .
Similarly we verify that , . Hence, we
have
. And, applying Property
(QL3) (see Assertion 2), we deliver . And,
taking into account that , , we see that
Condition (Preo2) is performed for and . That is
why, . But, since , then
. Thus, . Similarly, in the case where
(Preo3) is fulfilled for and , we successively
obtain:
1)
, , ; 2) ,
; 3)
, .
Thence, by Definition 5 (item (QL2)), we deduce,
. So, taking into account that ,
and , we obtain .
Let us prove that the relation is asymmetric,
i.e.:
(AS) If , and then
.
Indeed, suppose that and ,
where , ,
. Assume that . Then Condition
(Preo1) can not be performed for the elements and
. So, by conditions (Preo2), (Preo3), at least on of
the following two cases must hold:
[case 1] and
or
[case 2]
and .
But really each of these cases is impossible (by
definition of relation
). The contradiction obtained
above proves that .
Now, using the properties (AS) and (WT), we will
prove that the relation has the following property
of “stronged” asymmetry:
(AS(n)) If , and
then .
Indeed, let , and
. In the case the desired
result follows from Property (AS). In the case
we have . If we assume that
, we obtain . Thence, using
Property (AS) we obtain . Similarly we
get the equality in the cases
and . If we assume that
then we obtain . And, by Property
(WT), we deduce the impossible correlation .
Hence, the case is impossible.
And in all possible cases we obtain the desired result
for .
Now we consider any number such that
. Our inductive assumption is that Property
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(AS(k)) holds for all such that . Let
and . First
we assume that . Then we obtain
. Thence by Property
(WT), we deduce . So, by inductive
assumption, we have . The equality
together with the correlation, obtained before
leads to the correlation , which, by
Property (WT) leads to the contradiction .
Thus, the assumption is false. In
the case where or or
Property (AS(n)) can be reduced to Property
(AS(n-1)), which is valid, according to inductive
assumption. Therefore, the inductive transition is
well-founded. That is why, Property (AS(n)) holds for
each .
Let be a transitive closure (transitive hull) of
the relation in the sense of [18] (see page 337) or
[19], (see page 69), that is binary relation on
satisfying the following condition:
(PO) For the correlation is valid
if and only if there exist and
such, that , and .
The following inclusion holds:
(8)
Indeed, if assume that and , then
we can put , , . And, according
to (PO), we obtain .
It follows from the reflexivity of the relation
and the inclusion (8) that the relation is also
reflexive. According to Property (AS(n)) that the
relation is asymmetric (that is, if and
then ). Being a transitive closure of the relation
, the relation is transitive (according to [18] (see
Theorem and Definition 28.18) or [19], (see Theorem
5.7)). So if and then . Note that
the transitivity of the relation is not difficult to
check also by the direct verification method. Thus, the
relation is a partial order on . Therefore, by
Assertion 4, there exists a linear order relation
on such that . Then, using (8), we get the
inclusion:
(9)
Denote: . Also we define the mapping
by formula (5). That is for an arbitrary
we put . We
are going to prove that the mapping is an one-point
time on the oriented set .
1. According to formula (5), for any
we obtain:
where
Hence, the first condition of Definition 3 is satisfied.
2. Let , and
. Denote, , . Then for
elements and it is performed Condition
(Preo2). Therefore . Thence, by inclusion (9),
we deduce the inequality . And, since ,
we have . Moreover, by formula (5), we have
, . Hence, the second
condition of Definition 3 also is satisfied.
Thus, in accordance with Definition 3, the mapping
is a time on the oriented set .
3. Let’s prove that the time is one-point.
3.1. According to formula (5) the set consists of
one element. Hence, by Definition 4, the time is
quasi one-point.
3.2 Suppose that , and
. Then it follows from the quasi-one-pointness of
time that , . In
accordance with the formula (5), the last two
equalities are possible only if there exist numbers
such that ,
. Denote, , . Then,
according to the conditions (Preo1), (Preo2), taking
into account the reflexivity of the relation , we
obtain and . Hence, taking into
account the inclusion (9), we have and
(where , according to the above). That is
why:
(10)
Also, by formula (5), we obtain ,
. Now we are going to prove that
. Assume the the contrary, . Then, since the
oriented set is a quasi-chain, we deduce
.
Consequently, according to condition (Preo3), we
obtain and therefore . The last
inequality together with (10) ensures , which
is impossible, because , .
The obtained contradiction proves that .
Thus the both conditions of Definition 4 are
satisfied. Therefore the time is one-point.
4. Now we are going to prove that the time is
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-repeating. Using formula (5) for each
we obtain:
Rp
So, by Definition 6, the time is -repeating.
The lemma is completely proven.
Now Theorem 3 follows from Lemma 1 and Lemma
2.
In fact, applying Lemma 1 and Lemma 2 we can
readily deduce the following, more powerful theorem.
Theorem 4. For an oriented set the following
statements are equivalent:
(1) is a quasi-chain;
(2) admits an one-point time;
(3) admits -repeating one-point time.
7 On Images of Linearly Ordered
Sets
In this short section we deduce one interesting
corollary from Theorem 3 in the theory of ordered
sets. Namely it will be obtained the description of all
oriented sets, which can be represented as images of
linearly ordered sets. First of all we formulate the
definition of image of linearly ordered set.
Let be an oriented set and
be a mapping from to . Then we can
introduce the binary relation on the set
by the following rule:
For we note if and
only if there exist such, that
, and .
It is not difficult to verify that the ordered pair
is an oriented set, moreover
and
.
Definition 7. An oriented set is referred to as
image of the oriented set under the mapping
if and only if:
1. .
2. For the correlation
holds if and only if there exist such,
that , and
.
It is apparently that for each mapping
there exists an unique image under
the mapping . We will use the notation for
the image of the oriented set under the mapping
.
It is evidently that every linearly ordered set
is an oriented set with:
Therefore, it is meaningful to consider the image
of the linearly ordered set under some
mapping of kind . And the image of the
linearly ordered set is the oriented set . That
is why the following problem naturally arises:
Problem 2. Can an arbitrary oriented set be
represented as the image of some linearly
ordered set ? If it can not, describe all oriented sets
that can be represented as an image of some linearly
ordered set.
The key for solution of Problem 2 gives the
following Assertion.
Assertion 5. An oriented set can be
represented as image of some linearly ordered set if
and only if can be one-point chronologized.
Proof. Indeed, suppose that the ordered set
can be represented in the form , where
is a linearly ordered set. So, is the
mapping of kind with
. Here we denote by the binary relation,
inverse to (ie for the condition
holds if and only if ). According to Duality
Principle (see [20], page 14), the ordered pair:
(11)
is the linearly ordered set as well. It is not difficult to
verify that the mapping:
is an one-point time on (relatively the linearly
ordered set ). Conversely, let be a
linearly ordered set and be one-point
time on the oriented set . Then, by Definition 4, for
every time point the element
exists such, that . Consider the
mapping:
It is easy to verify that for this mapping it is
performed the equality , where the
linearly ordered set is determined by the formula
(11).
Assertion 5 together with Theorem 3 stipulate the
following corollary.
Corollary 1. An oriented set can be
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represented as image of some linearly ordered set if
and only if it is a quasi-chain.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Grushka Ya.I. is the only author in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
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