Representation of Steinitz's lattice in lattices of substructures of relational structures
General conditions under which certain relational structure contains a lattice of substructures isomorphic to Steinitz's lattice are formulated. Under some natural restrictions we consider relational structures with the lattice containing a sublattice isomorphic to the lattice of positive integ...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Representation of Steinitz's lattice in lattices of substructures of relational structures / O. Bezushchak, B. Oliynyk, V. Sushchansky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 184-201. — Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860062373742641152 |
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| author | Bezushchak, O. Oliynyk, B. Sushchansky, V. |
| author_facet | Bezushchak, O. Oliynyk, B. Sushchansky, V. |
| citation_txt | Representation of Steinitz's lattice in lattices of substructures of relational structures / O. Bezushchak, B. Oliynyk, V. Sushchansky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 184-201. — Бібліогр.: 19 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | General conditions under which certain relational structure contains a lattice of substructures isomorphic to Steinitz's lattice are formulated. Under some natural restrictions we consider relational structures with the lattice containing a sublattice isomorphic to the lattice of positive integers with respect to divisibility. We apply to this sublattice a construction that could be called ``lattice completion''. This construction can be used for different types of relational structures, in particular for universal algebras, graphs, metric spaces etc. Some examples are considered.
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| first_indexed | 2025-12-07T17:05:08Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 184–201
© Journal “Algebra and Discrete Mathematics”
Representation of Steinitz’s lattice in lattices
of substructures of relational structures
Oksana Bezushchak, Bogdana Oliynyk
and Vitaliy Sushchansky
Abstract. General conditions under which certain relational
structure contains a lattice of substructures isomorphic to Steinitz’s
lattice are formulated. Under some natural restrictions we consider
relational structures with the lattice containing a sublattice isomor-
phic to the lattice of positive integers with respect to divisibility. We
apply to this sublattice a construction that could be called “lattice
completion”. This construction can be used for different types of
relational structures, in particular for universal algebras, graphs,
metric spaces etc. Some examples are considered.
1. Introduction
Steinitz’s lattice was introduced at the beginning of the XX century by
German mathematician A.Steinitz for describing the structure of subfields
of algebraically closed field of prime characteristic [1]. It can be determined
as the lattice of supernatural numbers with a relation of the divisibility.
Steinitz’s lattice is complete, i.e. for an arbitrary subset of its elements
exists the exact lower and the exact upper bounds. It contains a various
sublattices including Boolean algebras. So, an existence of such lattice
in the lattice of substructures of mathematical structure shows its inner
richness that may be a basis for the use of the structure as an universal
object for corresponding class of structures of the same type.
2010 MSC: 03G10, 08A02, 03G05.
Key words and phrases: relational structure, lattice, supernatural numbers,
Boolean algebra.
O. Bezushchak, B. Oliynyk, V. Sushchansky 185
In the paper we formulate the general conditions in which certain
relational structure has some lattice of substructures, which is isomorphic
to Steinitz’s lattice. If these conditions are satisfied, then the isomorphic
representation of Steinitz’s lattice in a lattice of substructures of this
relational structure is obtained. Under certain natural restrictions it is
enough to view structures with the lattice containing sublattice, that is
isomorphic to the lattice of positive integers with divisibility. We apply
to this sublattice a construction that could be called “lattice completion”.
This construction can be used to different types of relational structures,
in particular — universal algebra, graphs, metric spaces etc. For example,
Steinitz’s lattice is isomorphic to:
(i) the lattice of all subfields of algebraic closure of finite field (see,
e.g., [2]);
(ii) the lattice of Glimm’s subalgebras of limited C∗-algebra (see,
e.g., [3], [4]);
(iii) lattices of so-called homogeneously symmetric and homogeneously
alternating subgroups of symmetric groups of permutations of nat-
ural numbers (see, e.g., [5], [6]).
Furthermore, some elements of Steinitz’s lattice of substructures in
certain relational structures were studied in many works of various authors
(see, e.g., [7]–[10]).
The paper is organized as follows. In Section 2 we give the definition
of Steinitz’s lattice and its basic properties. In Section 3 we present basic
information on the theory of relational structures. In Section 4 basic
construction of “lattice completion” of relational structures is described.
It is shown how to build the certain isomorphism between Steinitz’s
lattice and a lattice that occurs as a result of “lattice completion”. In
Section 5 we give examples of an application of the construction of “lattice
completion” in the theory of infinite groups and semigroups transforma-
tions. This makes it possible to introduce new objects namely, groups and
semigroups of periodically defined transformations of natural numbers.
Last section describes lattices of subspaces of Besicovitch’s space which
are constructed by the use of “lattice completion”, and therefore are
isomorphic to Steinitz’s lattice.
Results of the article were earlier partially announced in the article
[10] of third author. All symbols are commonly used in the paper. For
determination of indefinite terms we refer readers to [11]–[13].
186 Representation of Steinitz’s lattice
2. Steinitz’s lattice
2.1. Let N be a set of natural numbers and let P be its subset of primes.
Definition 1. Supernatural number (or Steinitz’s number) is called a
formal product ∏
p∈P
pkp , kp ∈ N ∪ {0, ∞}. (1)
Denote by the symbol SN the set of all supernatural numbers. Every
natural number is a supernatural number, so that N ⊂ SN. Numbers from
SN \N will be called infinite supernatural numbers. Divisible relation | on
N in natural way is extended to SN. Namely, for arbitrary supernatural
numbers
u =
∏
p∈P
pkp , v =
∏
p∈P
plp , kp, lp ∈ N ∪ {0, ∞}, (2)
we get u | v if and only if for all p ∈ P inequalities kp 6 lp hold (it
is assumed that ∞ is more than zero and all natural numbers). Main
properties of set SN, ordered by the divisible relation |, are characterized
by the following lemma.
Lemma 1. The ordered set (SN, |) is a lattice. The lattice (SN, |) is a
complete one and contains the largest and the smallest elements, that
accordingly are such supernatural numbers
I =
∏
p∈P
p∞ 1 =
∏
p∈P
p0. (3)
The proof of this statement is not difficult.
The exact lower and the exact upper bounds of supernatural numbers u,
v, that are given by their decompositions (2), are defined by the equalities
u ∨ v =
∏
p∈P
pmax(kp,lp) (4)
u ∧ v =
∏
p∈P
pmin(kp,lp) (5)
where max(k, ∞) = ∞, min(k, ∞) = k for k ∈ N ∪ {0}.
Definition 2. Lattice (SN, ∧, ∨) will be called Steinitz’s lattice.
The following lemma follows from equations (4), (5).
O. Bezushchak, B. Oliynyk, V. Sushchansky 187
Lemma 2. Steinitz’s lattice is a complete distributive lattice.
In the set of supernatural numbers we select two subsets.
Definition 3. Supernatural number u =
∏
p∈P pkp is called complete, if
for each p ∈ P there is inclusion kp ∈ {0, ∞}.
According to the definition a complete supernatural number u is
uniquely determined by the subset O(u) of that primes p from P, for
that kp = ∞. The set C of complete supernatural numbers is closed on
the operations ∨, ∧ and contains the numbers I and 1 that is defined
by (3). Moreover, on the set C one can define the operation of addition,
namely the addition ū of the number u ∈ C is called complete supernatural
number that is determined by subset O(ū) = P \ O(u). It is obvious, that
u ∨ ū = I, u ∧ ū = 1.
Lemma 3. The set C with the operations ∨, ∧, ¯ forms a Boolean algebra
with 1 as a zero element and I as an unit element. The Boolean algebra
(C, ∨, ∧, ¯ ) is isomorphic to the algebra of subsets of a countable set.
Proof. Define the mapping ϕ from the algebra of subsets of the set P to
the algebra C in such way. For any subset X ⊂ P put ϕ(X ) = u, were
O(u) = X . From mentioned above it follows that ϕ is a bijection. In
addition, for any X1, X2 ⊂ P for which ϕ(X1) = u1, ϕ(X2) = u2 we have
ϕ(X1 ∪ X2) = u1 ∨ u2, ϕ(X1 ∩ X2) = u1 ∧ u2, ϕ(X 1) = u1.
So, the mapping ϕ is an isomorphism, and this, in particular, means that
(C, ∨, ∧, ¯ ) is a Boolean algebra.
Definition 4. A supernatural number is called notsquare one, if indicators
of powers in its canonical decomposition (1) takes on only two values 0, 1.
Let B be a set of all notsquare supernatural numbers. It is clear, that
B is closed regarding to the operations ∨ and ∧. Moreover, on B can
also be defined the complement operation by the rule: for the number
u =
∏
p∈X p, we define
u =
∏
p∈P\X
p.
The number J =
∏
p∈P p is the largest element in the set B.
Lemma 4. Sublattice B of the lattice (SN, ∧, ∨) with the additional op-
eration, namely complement one, forms a Boolean algebra, the largest
element of which is number J, and the least element is 1. The algebra
(B, ∧, ∨, ¯ ) is isomorphic to the subalgebra of subsets of the set P.
188 Representation of Steinitz’s lattice
Proof. Define the mapping, which puts the subset X in correspondence
to the number
∏
p∈X p. This mapping is an isomorphism.
2.2. The sequence of positive integers χ = 〈k1, k2, . . .〉 will be called
divisible if ki | ki+1 for i = 1, 2, . . . . Let DS be a set of the most possible
of divisible sequences over N.
Definition 5. The sequence χ = 〈ki〉i∈N divides the sequence χ′ =
〈k′
i〉i∈N, if for any i ∈ N there are j ∈ N for which ki | k′
j .
Let | denote the divisibility of sequences. The relation | on DS is:
(i) reflexive, i.e. χ|χ for arbitrary sequence χ ∈ DS;
(ii) transitive, i.e. from χ1|χ2 and χ2|χ3 follows χ1|χ3 for any
χ1, χ2, χ3 ∈ DS.
But the relation of divisibility is not symmetric or antisymmetric relation.
Definition 6. Sequences χ and χ′ are called exactly divisible if at the
same time χ|χ′ and χ′|χ.
The exactly divisible relation is equivalence on DS, which we denote
by the symbol ∼. An arbitrary sequence χ ∈ DS determine a supernatural
number char χ (characteristic χ), which is defined thus
(i) each member of the sequence χ be a divisor of char χ;
(ii) every natural divisor of char χ be a divisor of some member of the
sequence χ.
For example, if χ = 〈1, p, p2, . . .〉, p ∈ P, then char χ = p∞, and when
χ = 〈1, 2!, 3!, 4!, . . .〉, then char χ = I. From the definition of characteristic
we get easy
Lemma 5. 1) For arbitrary χ1, χ2 ∈ DS the divisibility χ1|χ2 holds
if and only if char χ1| char χ2.
2) The sequences χ1, χ2 ∈ DS are exactly divisible if and only if when
char χ1 = char χ2.
So, sets of exactly divisible sequences are characterized by supernatural
numbers, moreover the correspondence between these classes of objects is
a bijective.
3. Relational structures
3.1. Recall that n-arity relation over the set A is called an arbitrary
subset of Cartesian degree An. A relational structure over the set A is
O. Bezushchak, B. Oliynyk, V. Sushchansky 189
called an a pair of the type
ℜ = 〈A, {Φkα
α }α∈I〉, (6)
where I is some set of indices, and for every α ∈ I Φkα
α is some relation
of the arity kα over A (kα ∈ N ∪ {0}). The set A is called a support of
the relational structure ℜ, the set {Φkα
α }α∈I is called its signature, and
the set (kα)α∈I is called its type.
Examples
1) Every directed graph without multiple edges with the set of vertices
V and the set of edges E ⊂ V × V is a relational structure (V, E) with
the support V and the signature E of the type (2).
2) Every colored graph with the set of vertices V , whose edges are
colored in k colors, is a relational structure with the support V and the
signature E1, . . ., Ek of the type (2, . . . , 2︸ ︷︷ ︸
k
).
3) Every metric space (X, d) with the set I of values of the metric is
a relational structure 〈X, {Dα}α∈I〉, where
Dα = {(x, y) | x, y ∈ X, d(x, y) = d(y, x) = α}.
This relational structure has the type (2α)α∈I .
4) Every universal algebra
£ = 〈A, {ϕkα
α }α∈I〉, where ϕkα
α : Akα → A
is an operation of the arity kα on A, can seen as a relational structure of
the form (6) of the type (kα + 1)α∈I , where
Φkα+1
α (x1, . . . , xkα
, y)
occurs if and only if, then ϕkα
α (x1, . . . , xkα
) = y.
The relational structures
ℜ = 〈A, {Φkα
α }α∈I〉 and ℜ′ = 〈B, {Ψ
lβ
β }β∈J〉
have the same type, if the sets I and J have the same cardinality and there
is a bijection f : I ↔ J so that for every α ∈ I the equality kα = lf(α)
holds.
190 Representation of Steinitz’s lattice
Let the relational structures ℜ and ℜ′ have the same type. The
bijection F : A → B is called an isomorphism of these structures, if for
any α ∈ I the relations
Φkα
α (x1, x2, . . . , xkα
), x1, x2, . . . , xkα
∈ A,
and Ψ
lf(α)
f(α) (F (x1), F (x2), . . . , F (xkα
))
are isomorphic.
In particular, an isomorphism of universal algebras or graphs as rela-
tional structures is their isomorphism in the usual sense, but an isomor-
phism of relational structures related to metric spaces means that these
spaces are isometric.
An isomorphism of relational structure itself is called an automor-
phism. All automorphisms of relational structure ℜ form a group with
the operation of superposition of automorphisms, which denoted by the
symbol Autℜ and named the group of automorphisms of the structure ℜ.
Let A′ be an arbitrary nonempty subset of the set A. For the subset
A′ we can consider restriction Φkα
α |A′ of the relation Φkα
α (α ∈ I) on the
set A′:
Φkα
α |A′(x1, x2, . . . , xkα
) = Φkα
α (x1, x2, . . . , xkα
) ∩ (A′)kα .
Note, that in this case some restrictions Φkα
α |A′ may be equal to empty
relations.
The relational structure 〈A′, {Φkα
α |A′}α∈I〉 is called a substructure of
the relational structure ℜ = 〈A, {Φkα
α }α∈I〉.
An isomorphism of ℜ onto some substructure of the structure ℜ′ is
called isomorphic emmbeding of the structure ℜ in the structure ℜ′.
3.2. The partially ordered set (I,>) is called a directed to the right, if
for any a, b ∈ I there exist such element c ∈ I, that a 6 c, b 6 c.
Definition 7. The family of structures {ℜi}i∈I and embedding fi,k :
ℜi → ℜk, i, k ∈ I, i 6 k, satisfying the following requirements:
1) I be a directed to the right partially ordered set;
2) for arbitrary indices i, k ∈ I, i 6 k, there exist emmbedings fi,k :
ℜi → ℜk, where fi,i be identity isomorphism;
3) if i, j, k ∈ I and i 6 j 6 k, then fi,j · fj,k = fi,k,
is called an inductive family Σ of relational structures over a set of
indices I.
O. Bezushchak, B. Oliynyk, V. Sushchansky 191
Every inductive family of relational structures
Σ = 〈ℜi, fi,k〉i,k∈I
defines the limit structure which is called inductive limit of family Σ and
is denoted by the symbol
ℜ(Σ) = lim
−−→
i
(ℜi, fi,k), i, k ∈ I. (7)
Elements of the structure (7) are the so-called strings, the relation Φkα
α
extends to their by the standard way ([13], pp. 151–156).
We will apply the construction of an inductive limit in a special case
when the index set I be the set of positive integers with the natural
order. In this case, the inductive family is the sequence ℜ1, ℜ2, . . ., and
morphisms fi,k will be defined as compositions of morphisms fi = fi,i+1
(i, k ∈ N). Moreover, sequences of the type u = akak+1 . . . (k ∈ N) are
strings, if the following conditions hold:
(i) ai ∈ Ai (i > k);
(ii) fi(ai) = ai+1 (i > k);
(iii) there is no an element ak−1 ∈ Ak−1, for which fk−1(ak−1) = ak.
Let (ui = a
(i)
li
a
(i)
li+1 . . . , 1 6 i 6 kα), and let Φkα
α be a relation from the
signature of relational structures ℜi, i ∈ N. From the definition follows
that the tuple of strings (u1, u2, . . . , ukα
) is in the relation Φkα
α if and
only if for l > max{l1, l2, . . . , lkα
} the tuples (a
(1)
l , a
(2)
l , . . . , a
(kα)
l ) is in
this relation. Let the subset ℜ(k) of the support of the structure ℜ(Σ),
be the set of all strings that was began with the elements from Al, l 6 k.
Then subset ℜ(k) determines a substructure of the structure ℜ(Σ), and
the equality
ℜ(Σ) =
∞⋃
k=1
ℜ(k)
takes place.
4. Construction of “lattice completion”
Let ℜ = 〈A, {Φkα
α }α∈I〉 be relational structures with the signature
(kα)α∈I . Suppose that for every natural number n the structure ℜ has
the only one its own substructure ℜ(n), moreover for the family of sub-
structures 〈ℜ(n), n ∈ N〉 following conditions hold:
(A) if n1 6= n2, then ℜ(n1) 6= ℜ(n2);
192 Representation of Steinitz’s lattice
(B) the inclusion ℜ(n1) ⊆ ℜ(n2) is satisfied if and only if, when n1|n2;
It follows from properties (A), (B) that family of substructures ℜ(n),
n ∈ N, forms a lattice under the inclusion. A correspondence n ↔ ℜ(n),
n ∈ N, is an isomorphism between this lattice and lattice (N, |) of natural
numbers.
Using the family ℜ(n), n ∈ N of substructures we define a new family
from ℜ as follows. Let χ = 〈n1, n2, . . .〉 ∈ DS be an arbitrary divisible
sequence of natural number. Construct now by sequence χ a growing
chain of substructures of structure ℜ of the form
ℜ(n1) ⊆ ℜ(n2) ⊆ · · · .
If sequence χ is bounded, then
ℜ(χ) =
∞⋃
i=1
ℜ(ni)
coincides with one of substructures ℜ(nk), k ∈ N. Therefore it suffices to
consider only divisible unbounded sequences. Suppose that for any χ as
unbounded divisible sequences the family of substructures ℜ(χ) satisfies
the following conditions:
(C) union ℜ(χ) is its own substructure of structure ℜ;
(D) ℜ(χ) does not coincide with any substructures ℜ(n), n ∈ N.
Definition 8. The family of substructures ℜ(χ), χ ∈ DS, of structure
ℜ will be called the lattice completion of lattice ℜ(n), n ∈ N.
Since DS contains arbitrary bounded divisible sequences, the family
ℜ(χ), χ ∈ DS, contains sublattice ℜ(n), n ∈ N.
Theorem 1. Suppose, that the family of substructures ℜ(n), n ∈ N, of
relational structure ℜ satisfies properties (A)–(D). Then
(i) substructures ℜ(χ1) and ℜ(χ2), χ1, χ2 ∈ DS, coincide if and only
if char χ1 = char χ2;
(ii) the family of substructures ℜ(χ), χ ∈ DS, forms a lattice under the
inclusion, which is isomorphic to Steinitz’s lattice.
Proof. (i) Suppose ℜ(χ1) and ℜ(χ2) are determined by divisible sequences
of natural numbers χ1 = 〈n
(1)
i 〉i∈N, χ2 = 〈n
(2)
i 〉i∈N. If ℜ(χ1) = ℜ(χ2),
then ℜ(χ1) ⊆ ℜ(χ2) and ℜ(χ2) ⊆ ℜ(χ1). The inclusion ℜ(χ1) ⊆ ℜ(χ2)
holds if and only if for any natural i there exists a number j, such
that ℜ(n
(1)
i ) ⊆ ℜ(n
(2)
j ). According to the property (B) it means that
O. Bezushchak, B. Oliynyk, V. Sushchansky 193
n
(1)
i |n
(2)
j , that is the sequence χ1 is a divisor of the sequence χ2. On
the other hand, the inclusion ℜ(χ2) ⊆ ℜ(χ1) means that for any j ∈ N
substructures ℜ(n
(2)
j ) is contained into some substructures ℜ(n
(1)
i ), namely
for an arbitrary j ∈ N there exists such i ∈ N, that n
(2)
j |n
(1)
i . This means
that the relation χ2|χ1 holds. Thus, sequences χ2 and χ1 are exactly
divisible. So, char χ1 = char χ2 by lemma 5.
Now suppose char χ1 = char χ2. Then sequences χ1 and χ2 are exactly
divisible, that is χ1|χ2 and χ2|χ1. As properties (A) and (B) hold for
the family ℜ(n), n ∈ N, using the considerations similar to above we
obtain that ℜ(χ1) ⊆ ℜ(χ2) and ℜ(χ2) ⊆ ℜ(χ1). In other words, these
substructures coincide.
(ii) Let DS(0) be a set of fixed representatives of each class of exactly
divisible sequences from DS. Then
{ℜ(χ) | χ ∈ DS} = {ℜ(χ) | χ ∈ DS(0)}.
We shall show, that the family of substructures on the right side of this
equality satisfies the condition (ii) of this theorem. Then the subset
of DS(0) is determined by classes of exactly divisible sequences on DS.
Hence, the mapping λ : SN → DS(0), such that λ(u) = χ iff char χ = u,
is a bijective by using lemma 5. So, properties (C), (D) of the family ℜ(χ),
χ ∈ DS, imply that a mapping λ̄ : DS(0) → {ℜ(χ) | χ ∈ DS} defined by
λ̄(χ) =
⋃
n∈χ
ℜ(n) = ℜ(χ), χ ∈ DS(0),
is a bijective too. Thus the mapping µ = λ · λ̄ : SN → {ℜ(χ) | χ ∈ DS(0)}
will also be a bijective. So that the image of arbitrary supernatural
number will be own substructure of the structure ℜ. It is also clear that
µ(1) = ℜ(1), µ(I) = ∪n∈χℜ(n), where χ be such divisible sequence, that
char χ = I. It remains to show that the mapping µ is consistent with the
lattice operations ∨ and ∧. To this end, we are going to check that µ is
consistent with the divisibility | on the set SN and the inclusion of ⊆ on
substructures {ℜ(χ) | χ ∈ DS(0)}. Indeed, let the condition u1|u2 hold
for supernatural numbers u1, u2. Then λ(u1)|λ(u2). It follows from (i)
that λ̄(λ(u1)) ⊆ λ̄(λ(u2)). So (λ · λ̄)(u1) ⊆ (λ · λ̄)(u2). The theorem is
proved.
Theorem 1 can be reformulated in terms of inductive limits as follows.
Let ℜ(n), n ∈ N be a family of structures. This family satisfies the
condition (A). There is an embedding ϕn1,n2 of the structure ℜ(n1) into
194 Representation of Steinitz’s lattice
the structure ℜ(n2) for arbitrary n1, n2 ∈ N, such that n1|n2. Assume
that for monomorphisms ϕn1,n2 following standard requirements:
(a) ϕn,n = Id for any n ∈ N;
(b) ϕn1,n3 = ϕn1,n2 · ϕn2,n3 for arbitrary n1, n2, n3 ∈ N such, that n1|n2
and n2|n3;
hold.
Then we have the direct spectrum 〈ℜ(n), ϕn,m〉n,m∈N of relational
structures, and let ℜ be a limit of this spectrum. For arbitrary sequence
χ = 〈ni〉i∈N it can be considered a direct spectrum 〈ℜ(ni), ϕni,ni+1〉i∈N
and its direct limit ℜ(χ). This structure can considered as a substructure
of structure ℜ in an obvious way.
Theorem 2. 1) Direct limits, defined by divisible sequences χ1 and χ2,
coincide as substructures of ℜ if and only if char χ1 = char χ2.
2) All boundary structures considered as substructures of ℜ and defined
by divisible sequences form a lattice with respect to inclusion. This
lattice is isomorphic to the Steinitz’s lattice.
We shall show examples, described how such construction works in
two different situations: for an universal algebras and metric spaces.
5. Groups and semigroups of periodically defined trans-
formations of natural numbers
Let PTn be a semigroup of all partial defined transformations of
set the {1, 2, . . . , n}, let PT (N) be a semigroup of all partial defined
transformations of the set of natural numbers N.
Definition 9. A transformation π̂ ∈ PT (N) is called periodically defined
expansion of the transformation π ∈ PTn on the set N, if its act on natural
numbers is defined by the following table
(
1 2 . . . n n + 1 n + 2 . . . 2n . . . . . .
1π 2π . . . nπ n + 1π n + 2π . . . n + nπ . . . . . .
)
.
A transformation α ∈ PT (N) is called periodically defined with the period
of definition n, if there is a transformation π ∈ PTn such, that α = π̂.
Every periodically defined transformation has infinitely many of dif-
ferent periods that are multiples of the minimal period. Let Tn, ISn, Sn
be, respectively, the complete semigroup of everywhere defined transfor-
mations of the set {1, 2, . . . , n}, the complete inverse semigroup of partial
O. Bezushchak, B. Oliynyk, V. Sushchansky 195
permutations and the complete symmetric group over this set. Thus
let T (N), IS(N), S(N) denote these semigroups over the set of natural
numbers.
Lemma 6. For an arbitrary n ∈ N the mapping ϕn : π → π̂, π ∈ PTn,
is a homomorphic emmbeding of the semigroup PTn into PT (N) and
its restrictions on Tn, ISn, Sn, respectively, is an emmbeding into T (N),
IS(N) and S(N).
Proof. The injection of the mapping ϕn follows directly from its definition
and its consistency with the operation of the multiplication of permuta-
tions is obvious. Moreover, for any partial permutation π ∈ ISn and its
inverse permutation π∗ we have ϕn(π∗) = π̂∗ = π̂∗. In particular, for any
everywhere defined permutation π ∈ Sn we shall get ϕn(π−1) = π̂−1. So,
ϕn will be a monomorphism ISn into IS(N) and Sn into S(N).
We will denote by the symbol Gn one of semigroups PTn, Tn, ISn, Sn,
n ∈ N, and will denote by the symbol G one of the corresponding semi-
groups PT (N), T (N), IS(N) or S(N). Thus, all formulated statements
will take place simultaneously for all four series of semigroups.
Let Ĝn be the image ϕn(Gn) of the semigroup Gn. Then Ĝn be a
subsemigroup in G, which is isomorphic to Gn. A family of semigroups
Ĝn, n ∈ N, in G is partially ordered by the inclusion. The main property
of this partially ordered set will be given in following lemma.
Lemma 7. A partially ordered set ({Ĝn, n ∈ N}, ⊆) is a lattice, which
is isomorphic to the lattice of positive integers with a relation of the
divisibility.
Proof. We shall verify that the correspondence n ↔ Ĝn, n ∈ N, will
be an isomorphism of partially ordered sets (N, |) and ({Ĝn, n ∈ N}, ⊆).
This correspondence is bijective, hence it is enough to show that for any
m, n ∈ N the condition m|n holds if and only if Ĝm ⊆ Ĝn. Let m|n and
n = km. So that the permutation π̂(k) given by the equality
π̂(k) =
(
1 2 . . . m . . . (k − 1)m + 1 . . . km
1π 2π . . . mπ . . . (k − 1)m + 1π . . . (k − 1)m + mπ
)
contains into Gn, and the equality holds:
ϕm(π) = ϕ(π̂(k)) = π̂.
So, for any transformation α ∈ Ĝ such, that α ∈ Ĝm, we obtain α ∈ Ĝn.
Hence, Ĝm ⊆ Ĝn.
196 Representation of Steinitz’s lattice
On the other hand, let Ĝm ⊆ Ĝn. In semigroup Ĝm there are permu-
tations with the minimal period of a definition m. Any such permutation
has the period of a definition n because it belongs to Ĝn. It follows that
m|n.
Since the conditions (A)–(B) from section 4 for the family {Gn, n ∈ N}
hold, then it is possible to apply the construction of lattice completion.
Namely, we introduce G(χ) by setting
G(χ) =
∞⋃
i=1
Ĝni
for an arbitrary sequence χ = 〈n1, n2, . . .〉 ∈ DS. Hence, for so determined
subsemigroups of the semigroup G the conditions (C) and (D) from
section 4 hold. This allows us to get the following result.
Theorem 3. The family of subsemigroups G(χ), χ ∈ DS, forms a lattice
regarding to an inclusion in the semigroup G, which is a Steinitz’s lattice.
Different elements of this lattice are pairwise non-isomorphic semigroups.
Proof. The first part of the statement is a direct consequence of Theorem 1.
So, we have to show its second part. Let first G = S(N) be the group of
permutations on the set N, and let G(u) = S(u) be the corresponding
group of periodically defined permutations (u ∈ SN). Due to [6] groups
G(u1) and G(u2) for u1, u2 ∈ SN, u1 6= u2, are non-isomorphic, because
one of them contains a permutation a centralizator that can be non
isomorphic to a centralizator of any permutation from another group.
Suppose now that G(u) is a subsemigroup of such periodically defined
permutations from G, that minimum periods are divisors of a supernatural
number u. Then the group G(u) of inverse elements coincides with S(u).
Hence, for u1 6= u2 semigroups G(u1) and G(u2) are non-isomorphic
because of groups of their inverse elements are non-isomorphic too.
The semigroup G(u), u ∈ SN, can be defined as a limit of the direct
spectrum of semigroups with so-called diagonal emmbeding. According to
[14] the emmbeding of the transitive transformations group (G, X) into the
transformations group (H, Y ) is called a diagonal emmbeding if orbits of
G onto the set Y either are trivial (consist of one point), or have the same
orbit cardinality |X|. Note, that the act G on such orbit is isomorphic to
the permutation group (G, X). A diagonal emmbeding of a group (G, X)
into a permutation group (H, Y ) is called a strictly diagonal emmbeding,
O. Bezushchak, B. Oliynyk, V. Sushchansky 197
if there are no trivial orbits of group G onto Y . In this case, with some
k ∈ N we have |Y | = k|X|. Note that strictly diagonal emmbeding can
be defined for arbitrary, not necessarily transitive permutations groups.
So, its can be defined for transformations semigroups too.
Definition 10. An emmbeding of a transformation semigroup (V, X)
into a transformation semigroup (W, Y ) will be called (strictly) diagonal
emmbeding, if there is a partition of Y onto subsets of the capacity |X|,
which are invariant under the action of the image of V . Moreover, the
action of the image of V onto each of these subsets is isomorphic as a
semigroup of transformations to semigroup (V, X).
As before, let Gn ∈ {PTn, Tn, ISn, Sn}.
Lemma 8. Let the mapping δk : Gn → Gnk is defined by equality
δk(π) = π̂(k),
for any permutation π ∈ Gn. Then δk is an isomorphic emmbeding of the
semigroup Gn into the semigroup Gnk.
A proof is done by a direct check as at lemma 6.
For an arbitrary divisible sequence χ = 〈n1, n2, . . .〉 ∈ SN, ni+1/ni = ki
(i = 1, 2, . . .) we define a direct spectrum of semigroups
Gni
−→δk1 Gn2 −→δk2 Gn3 −→ · · · .
So, we also have to consider the limit semigroup of the spectrum
G[χ] = lim
→i
(Gni
, δki
).
Theorem 4. For any supernatural number χ ∈ SN semigroups G[χ] and
G(χ) are isomorphic as semigroups of transformations.
Proof. An isomorphism of semigroups is constructed in the standard way,
and the set of an action of G[χ] is naturally identified with N. Finally, we
note that our semigroups act equally on the set N.
6. Steinitz’s lattice of subspaces in Besicovitch’s space
A normalized Hamming metric on the set Hn of (0, 1)-sequences of a
length n is called a metric dH , defined by the following equality
dH(x, y) =
1
n
n∑
i=1
|xi − yi|, x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Hn. (8)
198 Representation of Steinitz’s lattice
So, let now {0, 1}N be a set of infinite (0, 1)-sequences. We will represent
the distance function d̂B by the rule
d̂B(x, y) = lim
n→∞
sup dH((x1, . . . , xn), (y1, . . . , yn)), (9)
x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ {0, 1}n.
Since (8) is a metric, it is easy to verify that the function d̂B, defined
by (9), is a pseudometric, i.e., it differs from a metric so that there are
infinite sequences with the distance between them equals to 0. The binary
relation
x ∼B y ⇔ d̂B(x, y) = 0
is an equivalence on {0, 1}N. Define
XB = {0, 1}N/∼B
.
The function d̂B is consistent with the equivalence ∼B. Hence, it deter-
mines a function dB on XB as follows:
dB([x], [y]) = d̂B(x, y), (10)
where [x], [y] are equivalence classes of ∼B, and x ∈ [x], y ∈ [y] are
arbitrary representatives of these classes. Defined by the equality (10) the
function dB is a metric. So, a metric space (XB, dB) is called Besicovitch’s
space (see [15]).
For any natural number n normalized Hamming space Hn is isometric
embedded into Besicovitch’s space XB.
Lemma 9. The mapping hn : Hn → XB, defined by setting
hn((x1, . . . , xn)) = [(x1, . . . , xn, x1, . . . , xn, . . .)], (x1, . . . , xn) ∈ Hn,
is an isometric emmbeding.
Proof. By the definition the distance (10) between classes of the equiva-
lence ∼B, which defined by periodic sequences
x = x1, . . . , xn, x1, . . . , xn, . . . and y = y1, . . . , yn, y1, . . . , yn, . . . ,
is equal to
1
n
dH((x1, . . . , xn), (y1, . . . , yn)).
O. Bezushchak, B. Oliynyk, V. Sushchansky 199
Denote by H̃n the image of Hamming space Hn for the mapping hn.
Lemma 10. The inclusion H̃n ⊆ H̃m (n, m ∈ N) holds if and only if
then n|m.
Proof. It is obvious.
So that, Besicovitch’s space XB contains the family of subspaces H̃n,
n ∈ N. It is easy to see, that for this family conditions (A) − (D) of the
lattice completion hold. It follows from Theorem 1, that the space XB
contains a family of subspaces H̃u, u ∈ SN, indexing by supernatural
numbers. Note, that the subspace H̃u consists of all possible periodic
(0, 1)-sequences, such that lengthes of their minimum periods are divisors
of a supernatural number u. By properties of the general construction we
obtain the following
Theorem 5. A family of subspaces H̃u, u ∈ SN, of the space XB forms a
lattice over the inclusion which is isomorphic to the lattice of supernatural
numbers. If u1 6= u2, then subspaces H̃u1 and H̃u2 are not isometric.
Proof. The first part of the assertion follows from Theorem 1. The proof
of the third part given in the article [16].
By the Theorem 1 each of spaces H̃u, u ∈ SN \ N, is isometric to
inductive limit of the sequence of finite Hamming spaces Hm1 , Hm2 ,
. . ., where χ = 〈m1, m2, . . .〉 is such divisible sequence that char χ = u.
Monomorphisms are the diagonal emmbedings fi : Hmi
→ Hmi+1 , defined
by following equalities
fi((x1, . . . , xmi
)) = (x1, . . . , xmi
, . . . , x1, . . . , xmi︸ ︷︷ ︸
kimi
), (x1, . . . , xmi
) ∈ Hmi
,
(11)
where ki = mi+1/mi, i = 1, 2, . . .. Note, that a construction of limit cube,
that is built as inductive limit of the sequence of Hamming spaces H2i of
the dimension 2i with emmbedings of following doubling coordinates:
δi((x1, . . . , x2i)) = (x1, x1, x2, x2, . . . , x2i , x2i), i = 1, 2, . . . ,
is considered in [17]. So that, it is easy to understand that the limit space
of such direct spectrum is isometric to the space H̃2∞ with emmbedings
of the type (11). Note, that in [18] was considered continuum family
of subspaces of the Besicovitch space on some alphabet B, naturally
parametrized by supernatural numbers. Every subspace is defined as
200 Representation of Steinitz’s lattice
a diagonal limit of finite Hamming spaces on the alphabet B. So, our
construction is more general.
Other generalizations of this construction are proposed in [19], and a
generalization of construction of lattice completion for linear groups is
considered in the article [14].
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and application. Chernivtsi, Ruta, 2003, (Ukrainian).
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[16] B. V. Oliynyk, V. I. Sushchanskii The isometry groups of Hamming spaces of
periodic sequences, Siberian Mathematical Journal, Vol. 54, N. 1, 2013, pp. 124–136.
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N. 14, 2008, pp. 1454–1461.
O. Bezushchak, B. Oliynyk, V. Sushchansky 201
[18] B. Oliynyk, The diagonal limits of Hamming spaces, Algebra and Discrete
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Contact information
O. Bezushchak Department of Mechanics and Mathematics
Kyiv National Taras Shevchenko University
Volodymyrska, 64, Kyiv 01033, Ukraine
E-Mail(s): bezusch@univ.kiev.ua
B. Oliynyk Department of Mathematics, National Univer-
sity of Kyiv-Mohyla Academy, Skovorody St. 2,
Kyiv, 04655, Ukraine
E-Mail(s): oliynyk@ukma.edu.ua
V. Sushchansky Institute of Mathematics Silesian University of
Technology ul. Kaszubska 23, 44-100 Gliwice,
Poland
E-Mail(s): Vitaliy.Sushchanskyy@polsl.pl
Received by the editors: 11.05.2016.
|
| id | nasplib_isofts_kiev_ua-123456789-155237 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T17:05:08Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Bezushchak, O. Oliynyk, B. Sushchansky, V. 2019-06-16T14:26:24Z 2019-06-16T14:26:24Z 2016 Representation of Steinitz's lattice in lattices of substructures of relational structures / O. Bezushchak, B. Oliynyk, V. Sushchansky // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 184-201. — Бібліогр.: 19 назв. — англ. 1726-3255 2010 MSC:03G10, 08A02, 03G05. https://nasplib.isofts.kiev.ua/handle/123456789/155237 General conditions under which certain relational structure contains a lattice of substructures isomorphic to Steinitz's lattice are formulated. Under some natural restrictions we consider relational structures with the lattice containing a sublattice isomorphic to the lattice of positive integers with respect to divisibility. We apply to this sublattice a construction that could be called ``lattice completion''. This construction can be used for different types of relational structures, in particular for universal algebras, graphs, metric spaces etc. Some examples are considered. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Representation of Steinitz's lattice in lattices of substructures of relational structures Article published earlier |
| spellingShingle | Representation of Steinitz's lattice in lattices of substructures of relational structures Bezushchak, O. Oliynyk, B. Sushchansky, V. |
| title | Representation of Steinitz's lattice in lattices of substructures of relational structures |
| title_full | Representation of Steinitz's lattice in lattices of substructures of relational structures |
| title_fullStr | Representation of Steinitz's lattice in lattices of substructures of relational structures |
| title_full_unstemmed | Representation of Steinitz's lattice in lattices of substructures of relational structures |
| title_short | Representation of Steinitz's lattice in lattices of substructures of relational structures |
| title_sort | representation of steinitz's lattice in lattices of substructures of relational structures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155237 |
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