# Algebraic closures and their variations111The work was carried out in the
framework of the State Contract of the Sobolev Institute of Mathematics,
Project No. FWNF-2022-0012, and partially supported by Committee of Science in
Education and Science Ministry of the Republic of Kazakhstan, Grant No.
AP19677451.

Sergey V. Sudoplatov

###### Abstract

We study possibilities for algebraic closures, differences between definable
and algebraic closures in first-order structures, and variations of these
closures with respect to the bounds of cardinalities of definable sets and
given sets of formulae. Characteristics for these possibilities and
differences are introduced and described. These characteristics are studied
for some natural classes of theories. Besides algebraic closure operators with
respect to sets of formulae are introduced and studied. Semilattices and
lattices for families of these operators are introduced and characteristics of
these structures are described.

Key words: algebraic closure, definable closure, degree of algebraization,
algebraic set, $\Delta$-algebraic element, semilattice.

##  1 Introduction

The notions of definable and algebraic closures are broadly used in Model
Theory and applications [1, 2, 3, 4, 5].

In the paper, we study possibilities for algebraic closures, differences
between definable and algebraic closures in first-order structures, and
variations of these closures with respect to the bounds of cardinalities of
definable sets and given sets of formulae. Characteristics for these
possibilities and differences are introduced and described. These
characteristics are studied for some natural classes of theories.

The paper is organized as follows. In Section 2, preliminary notions,
notations and properties for algebraic and definable closures are considered.
In Section 3, we study variations of algebraic closure, properties and
possibilities of these variations, of degree of algebraization, and of
difference between definable and algebraic closures, illustrating these
possibilities by a series of examples. Algebraic sets and their degrees are
studied in Section 4. In Section 5, we consider algebraic closures relative
sets of formulae, study a hierarchy of operators of algebraic closures
relative these sets. We introduce semilattices and lattices for families of
these operators and describe some characteristics of these structures.

Throughout we use both the standard model-theoretic terminology and notions of
Lattice Theory [6, 7, 8].

##  2 Pregeometries and closures

Definition [2, 3, 4, 5], cf. [9]. For a set $S$, its Boolean $P(S)$, and an
operator ${\rm cl}\mbox{: }P(S)\to P(S)$, a pair $\langle S,{\rm cl}\rangle$
is called a pregeometry or a matroid, if it satisfies the following conditions
for any $X\subseteq S$ and $a,b\in S$:

(i) $X\subseteq{\rm cl}(X)$ (Reflexivity);

(ii) ${\rm cl}({\rm cl}(X))={\rm cl}(X)$ (Transitivity);

(iii) if $a\in{\rm cl}(X)$ then $a\in{\rm cl}(Y)$ for some finite $Y\subseteq
X$ (Finite character).

(iv) if $a\in{\rm cl}(X\cup\\{b\\})\setminus{\rm cl}(X)$ then $b\in{\rm
cl}(X\cup\\{a\\})$ (Exchange property);

A pregeometry $\langle S,{\rm cl}\rangle$ is called a geometry if:

(v) ${\rm cl}(\emptyset)=\emptyset$, and for every $a\in S$, ${\rm
cl}(\\{a\\})=\\{a\\}$ (ES-property).

The operator ${\rm cl}$ for the pregeometry is called the closure operator. By
the definition the closure operator for a pregeometry produces a geometry if
the closure of the empty set is again empty and the closure of any singleton
is again a singleton, which, by reflexivity, equals that singleton.

###### Remark 2.1

[3]. Any pregeometry $\langle S,{\rm cl}\rangle$ produces a canonical geometry
$\langle S^{\prime},{\rm cl}^{\prime}\rangle$ putting $S^{\prime}=\\{{\rm
cl}(\\{a\\})\mid a\in S\setminus{\rm cl}(\emptyset)\\}$ and for $X\subseteq
S$, ${\rm cl}^{\prime}(\\{{\rm cl}(\\{a\\})\mid a\in X\\})=\\{{\rm cl}(b)\mid
b\in{\rm cl}(X)\\}$.

In fact the ES-property for ${\rm cl}^{\prime}$ is implies by the reflexivity
and transitivity for the operator ${\rm cl}$.

We fix a big saturated structure $\mathcal{M}$ and its theory $T={\rm
Th}(\mathcal{M})$. Following [10] by a set $A$ of $T$ we mean a subset of the
universe $M$ in the structure $\mathcal{M}$ satisfying some type ${\rm
tp}(A,\mathcal{M})$. Similarly, a tuple $\overline{a}$ in $T$ is a tuple of
elements in $\mathcal{M}$ satisfying given type ${\rm tp}(\overline{a})$.

In [1], two types of closures in structures are considered, algebraic and
definable, as well as the following concepts related to them:

Definition. [1, 2, 5] 1\. The tuple $\overline{b}$ is defined by the formula
$\varphi(\overline{x},\overline{a})$ of $T$ with parameters $\overline{a}$, if
$\varphi(\overline{x},\overline{a})$ has unique solution $\overline{b}$.

The tuple $\overline{b}$ is defined by the type $p$ if $\overline{b}$ is the
unique tuple which realizes $p$. It is definable over a set $A$ if ${\rm
tp}(\overline{b}/A)$ defines it.

2\. For a set $A$ of a theory $T$ the union of sets of solutions of formulae
$\varphi(x,\overline{a})$, $\overline{a}\in A$, such that
$\models\exists^{=n}x\>\varphi(x,\overline{a})$ for some $n\in\omega$
(respectively $\models$ $\exists^{=1}x\>\varphi(x,\overline{a})$) is said to
be an _algebraic(definable or definitional) closure_ of $A$. An algebraic
closure of $A$ is denoted by ${\rm acl}(A)$ and its definable (definitional)
closure, by ${\rm dcl}(A)$.

In such a case we say that the formulae $\varphi(x,\overline{a})$ witness that
algebraic / definable (definitional) closure, and these formulae are called
algebraic / defining.

Any element $b\in{\rm acl}(A)$ (respectively, $b\in{\rm dcl}(A)$) is called
algebraic (definable or definitional) over $A$. If the set $A$ is fixed or
empty, we just say that $b$ is algebraic, (definable, or definitional).

3\. If ${\rm dcl}(A)={\rm acl}(A)$, ${\rm cl}_{1}(A)$ denotes their common
value.

4\. If $A={\rm acl}(A)$ (respectively, $A={\rm dcl}(A)$ ) then $A$ is called
algebraically (definably) closed.

5\. The type $p$ is algebraic (defining) if it is realized by finitely many
tuples (unique one) only, i.e., it contains an algebraic (defining) formula
$\varphi$. This formula $\varphi$ can be chosen with the minimal number of
solution, and in such a case $\varphi$ isolates $p$. The number of these
solutions is called the degree ${\rm deg}(p)$ of $p$.

6\. The complete algebraic types $p(x)\in S(A)$ are exactly ones of the form
${\rm tp}(a/A)$, where $a$ is algebraic over $A$. The degree of $a$ over $A$,
${\rm deg}(a/A)$ is the degree of ${\rm tp}(a/A)$.

###### Remark 2.2

[1]. The pairs $\langle M,{\rm acl}\rangle$ and $\langle M,{\rm dcl}\rangle$
satisfy the following properties:

(i) the reflexivity: it is witnessed by the formula $x\approx y$;

(ii) the transitivity: if the formulae
$\varphi_{1}(x_{1},\overline{a}),\ldots,\varphi_{n}(x_{n},\overline{a})$
witnessed that $b_{1},\ldots,b_{n}\in{\rm acl}(A)$ (respectively,
$b_{1},\ldots,b_{n}\in{\rm dcl}(A)$) and the formula
$\psi(x,b_{1},\ldots,b_{n})$ witnesses that $c\in{\rm
acl}(\\{b_{1},\ldots,b_{n}\\})$ (respectively, $c\in{\rm
dcl}(\\{b_{1},\ldots,b_{n}\\})$) then the formula

$\exists
x_{1},\ldots,x_{n}\left(\psi(x,x_{1},\ldots,x_{n})\wedge\bigwedge\limits_{i=1}^{n}\varphi_{i}(x_{i},\overline{a})\right)$
(1)

witnesses that $c\in{\rm acl}({\rm acl}(A))$ (respectively, $c\in{\rm
dcl}({\rm dcl}(A))$);

(iii) the finite character: if a formula $\varphi{x,\overline{a}}$ witnesses
that $a\in{\rm acl}(A)$ (respectively, $a\in{\rm dcl}(A)$) then $a\in{\rm
acl}(A_{0})$ for the finite $A_{0}\subseteq A$ consisting of coordinates in
$\overline{a}$.

##  3 $n$-Closures and degrees of algebraization

Now we consider modifications of the algebraic closure related to the sets of
solutions of formulas bounded in cardinality by some natural number.

Definition. 1\. For $n\in\omega\setminus\\{0\\}$ and a set $A$ an element $b$
is called $n$-algebraic over $A$, if $a\in{\rm acl}(A)$ and it is witnessed by
a formula $\varphi(x,\overline{a})$, for $\overline{a}\in A$, with at most $n$
solutions.

2\. The set of all $n$-algebraic elements over $A$ is denoted by ${\rm
acl}_{n}(A)$.

3\. If $A={\rm acl}_{n}(A)$ then $A$ is called $n$-algebraically closed.

4\. The type $p$ is $n$-algebraic if it is realized by at most $n$ tuples
only, i.e., ${\rm deg}(p)\leq n$.

5\. The complete $n$-algebraic types $p(x)\in S(A)$ are exactly ones of the
form ${\rm tp}(a/A)$, where $a$ is $n$-algebraic over $A$, i.e., with ${\rm
deg}(a/A)\leq n$. Here ${\rm deg}(a/A)=k\leq n$ defines the $n$-degree ${\rm
deg}_{n}(a/A)$ of ${\rm tp}(a/A)$ and of $a$ over $A$.

6\. If ${\rm acl}(A)={\rm acl}_{n}(A)$ then minimal such $n$ is called the
degree of algebraization over the set $A$ and it is denoted by ${\rm deg}_{\rm
acl}(A)$. If that $n$ does not exist then we put ${\rm deg}_{\rm
acl}(A)=\infty$. The supremum of values ${\rm deg}_{\rm acl}(A)$ with respect
to all sets $A$ of given theory $T$ is denoted by ${\rm deg}_{\rm acl}(T)$ and
called the degree of algebraization of the theory $T$.

7\. Following [11] theories $T$ with ${\rm deg}_{\rm acl}(T)=1$, i.e., with
defined ${\rm cl}_{1}(A)$ for any set $A$ of $T$, are called quasi-Urbanik,
and the models $\mathcal{M}$ of $T$ are quasi-Urbanik, too.

###### Remark 3.1

By the definition any algebraic type is $n$-algebraic for some $n$, it is
isolated and have a fixed finite set of solutions in any elementary extension
of given model. In particular, definable types are $1$-algebraic and have
unique realizations in any given model. We have ${\rm dcl}(A)={\rm
acl}_{1}(A)$ for any $A$.

Notice that any set $B$ of realizations of an algebraic type over $A$ is the
union of the subsets $B_{n}$ of realizations for its $n$-algebraic subtypes
for all $n$: $A=\bigcup\limits_{n\in\omega}B_{n}$. Moreover, the sets $B_{n}$
form an increasing chain, with $B_{m}=B_{n}$ for $m>n$.

Notice also that it is essential in the definition of $n$-algebraic types that
models for the sets of realizations are arbitrary, since, for instance, a
theory $T$ of infinitely many disjoint nonempty unary predicates $U_{i}$,
$i\in I$, has models with arbitrarily finitely many realizations of
nonisolated type $p(x)=\\{\neg U_{i}(x)\mid i\in I\\}$. This type has
infinitely many realizations in an appropriate $T$-model, and therefore it is
not algebraic.

The following proposition allows to transform algebraic types into their
finite subtypes.

###### Proposition 3.2

(cf. [1, Lemma 6.1]). $1.$ A type $p$ defines a tuple if and only if for some
finite $q\subseteq p$, $\bigwedge q$ defines that tuple.

$2.$ A type $p$ is $n$-algebraic if and only if for some finite $q\subseteq
p$, $\bigwedge q$ is $n$-algebraic; moreover, $q$ can be chosen with the same
set of realizations as for $p$.

Recall [2] that for a set $A$ and an element $a$ the $A$-orbit ${\rm
Orb}_{A}(a)$ of $a$ is the set of all elements $b$ in the given structure
which are connected with $a$ by an $A$-automorphism.

The following proposition gives an algebraic characterization for
$n$-algebraic types.

###### Proposition 3.3

A type $p$ is $n$-algebraic over $A$ if and only if any / some
$(|A|+|T|)$-saturated model $\mathcal{M}$ containing $A$ has finitely many
$A$-orbits $O$ consisting of realizations of $p$, all these orbits are finite,
and moreover the union $\bigcup O$ has at most $n$-elements. If $p$ is
complete then that $A$-orbit is unique in $\mathcal{M}$.

Proof. Taking a $(|A|+|T|)$-saturated model $\mathcal{M}$ with $A\subseteq M$
and a $n$-algebraic type $p$ over $A$ we have finitely many possibilities for
links, by $A$-automorphisms, between realizations of $p$. Moreover, if
$b_{1},b_{2},\ldots,b_{n}$ are all realizations of $p$ then for any $b_{i}$,
$b_{j}$ either ${\rm tp}(b_{i}/A)\neq{\rm tp}(b_{j}/A)$ and $b_{i}$, $b_{j}$
belong to distinct $A$-orbits, or ${\rm tp}(b_{i}/A)={\rm tp}(b_{j}/A)$ and
$b_{i}$, $b_{j}$ belong to a common $A$-orbit connected by an $A$-automorphism
$f$ with $f(b_{i})=f(b_{j})$. Additionally, if $c$ is not a realization of $p$
then $c$ does not belong to orbits of elements $b_{1},b_{2},\ldots,b_{n}$.

Conversely, if any / some $(|A|+|T|)$-saturated model $\mathcal{M}$ containing
$A$ has finitely many $A$-orbits $O$ consisting of realizations of $p$ and the
union $\bigcup O$ has at most $n$-elements then by the saturation of
$\mathcal{M}$ the type $p$ can not have more than $n$ realizations implying
that $p$ is $n$-algebraic over $A$.

###### Remark 3.4

Taking cycles of arbitrarily large diameters we can find an algebraic type
with arbitrary large and arbitrary many finite orbits. At the same time, it is
essential in Proposition 3.3 that the model is saturated, since following
Remark 3.1 there are non-algebraic types with arbitrarily many realizations,
and taking copies of Example in Remark 3.1 we can obtain arbitrarily many
finite orbits.

Notice also that the properties of $n$-algebraicity over a set $A$ are both
syntactic and semantic, since on the one hand it is written by a family of
formulae $\varphi(x,\overline{a})\wedge\exists^{\leq
n}x\varphi(x,\overline{a})$, for $\overline{a}\in A$, and on the other hand it
is checked in any model $\mathcal{M}$ of given theory $T$ with $A\subseteq M$
and satisfying the given type ${\rm tp}(A)$.

For instance, taking a structure $\mathcal{M}$ with $A\subseteq M$ such that
each element of $M$ is the unique solution of an appropriate formula
$\varphi(x,\overline{a})$, for $\overline{a}\in A$, then $M$ is definably
closed, i.e., $1$-algebraically closed, with unique automorphism $f$ fixing
$A$ pointwise and this automorphism is identical. Following [2] the structure
$\mathcal{M}$ with unique automorphism is called rigid. Thus the
$1$-algebraicity of each element in $M$ over the empty set, i.e., the
condition $M={\rm dcl}(\emptyset)$ produces the rigid structure $\mathcal{M}$.

We separate the forms of rigidity of a structure as follows: the semantic one
is defined in terms of trivial automorphism group, and the syntactic one is in
terms of $1$-algebraicity of the universe over the empty set.

Thus, any syntactically rigid structure is semantically rigid and it is
defined uniquely. But not vice versa, in general. Indeed, taking a structure
$\mathcal{M}$ consisting of infinitely many distinct constants $c_{i}$, $i\in
I$, we have both the semantic and syntactic rigidity. Extending $\mathcal{M}$
by a single realization of the nonisolated type $p(x)=\\{\neg x\approx
c_{i}\mid i\in I\\}$ we obtain a structure $\mathcal{N}\succ\mathcal{M}$
preserving the semantic rigidity but loosing the syntactic one. Taking
disjoint unions [12] of copies of $\mathcal{M}$ one can obtain semantically
rigid elementary extensions with arbitrarily many new elements. Hence, in
appropriate cases, the semantic rigidity can be transformed for elementary
extensions.

These appropriate cases admits that one-element finite orbits are preserved
under elementary extensions. We observe this preservation for structures whose
universes consist of constants only. And one-element finite orbits are not
preserved if, for instance, a structure consists of infinitely many two-
element equivalence classes expanded by constants for all elements of these
classes.

Similarly, following Proposition 3.3 homogeneous structures $\mathcal{M}$ with
at most $n$-element finite orbits, under appropriate additional conditions,
produce the $n$-algebraicity. And some opposite conditions imply the negation
of $n$-algebraicity for $\mathcal{N}\equiv\mathcal{M}$.

Thus, in general case, poor automorphism groups, in particular, identical
automorphism groups ${\rm Aut}(\mathcal{M})$ can not reflect adequately links
for algebraic closures in structures $\mathcal{N}\equiv\mathcal{M}$.

Recall that a theory $T$ is $n$-transitive, for $n\in\omega$, if the type
${\rm tp}(\overline{a})$ of each tuple $\overline{a}$ with $l(\overline{a})=n$
is forced by formulae describing that coordinates of $\overline{a}$ coincide
or do not coincide.

###### Remark 3.5

Clearly, any $n$-transitive theory is $m$-transitive for every $m\leq n$.
Besides, algebraic characteristics for $n$-transitive theories such as degrees
of algebraicity are really defined by sets $A$ of cardinalities $\geq n$,
since ${\rm acl}(B)=B$ for all smaller sets $B$.

Thus, in general case, structures for ${\rm acl}(B)$ with small finite
cardinalities $|B|$ does not reflect general behavior of the algebraic closure
operator. It is valid for the operators ${\rm acl}_{n}(\cdot)$, too.

###### Remark 3.6

In view of Remarks 3.4 and 3.5, nir poor automorphism groups, for structure
similar to rigid, nor rich automorphism groups, for structure similar to
transitive, do not reflect the frame of given structures and the frame for
their theories in the level of algebraic closure.

###### Proposition 3.7

(cf. [1, Lemma 6.2]). $1.$ $A\subseteq{\rm acl}_{m}(A)\subseteq{\rm
acl}_{n}(A)\subseteq{\rm acl}(A)$, for any $0<m\leq n$. In particular,
$\langle M,{\rm acl}_{n}\rangle$ is reflexive for any $n$.

$2.$ If $A\subseteq B$ and $n\geq 1$ then ${\rm acl}_{n}(A)\subseteq{\rm
acl}_{n}(B)$.

$3.$ If $A$ is definably (algebraically) closed then $A={\rm dcl}(A)$ ($A={\rm
acl}(A)$).

$4.$ If $A$ is $n$-algebraically closed for $n\geq 2$ then $A={\rm acl}(A)$
iff any finite orbit over $A$ has at most $n$ elements.

$5.$ A tuple $\overline{b}$ is definable (algebraic) over $A$ if and only if
$\overline{b}\in{\rm dcl}(A)$ ($\overline{b}\in{\rm acl}(A)$).

Proof. Items 1, 2, 3, 5 immediately follow by the definition and [1, Lemma
6.2]. Item 4 is implied by Proposition 3.3.

Proposition 3.7 immediately implies:

###### Corollary 3.8

If ${\rm cl}_{1}(A)$ exists then it equals ${\rm acl}_{n}(A)$ and it is
$n$-algebraically closed for each $n$.

###### Corollary 3.9

For any structure $\mathcal{M}$, the pairs $\mathcal{S}_{n}=\langle M,{\rm
acl}_{n}\rangle$, $n\in\omega$, define ascending chains $({\rm
acl}_{n}(A))_{n\in\omega}$ for each $A\subseteq M$, where ${\rm
acl}_{0}(A)=A$. These pairs $S_{n}$ coincide starting with some $n_{0}$ iff
all chains $({\rm acl}_{n}(A))_{n\geq n_{0}}$ are singletons, and iff any
finite orbit over arbitrary $A\subseteq M$ has at most $n_{0}$ elements in a
homogeneous elementary extension of $\mathcal{M}$.

###### Remark 3.10

Since for $k\leq m$ there are $A^{k}_{m}=\frac{m!}{(m-k)!}$ $k$-tuples
consisting of pairwise distinct elements in the given $m$-element set we can
not assert Item 5 of Proposition 3.7 for $n$-algebraic tuples. Indeed, orbits
for tuples can be much more greater than orbits for elements. In particular,
it holds for structures in the empty language.

###### Remark 3.11

Notice that along with the reflexivity for ${\rm acl}_{n}$, by the definition
any operator ${\rm acl}_{n}$ has the finite character. At the same time ${\rm
acl}_{n}$ can be non-transitive. Indeed, taking an element $a$ and ${\rm
acl}_{n}({\rm acl}_{n}(\\{a\\}))$ one can find a structure $\mathcal{M}$, say
a graph, with $n$ new elements $b_{1},\ldots,b_{n}$ as solutions of
$n$-algebraic formula $\varphi(x,a)$ isolating an $n$-algebraic type $p(x)$
such that the $n$-algebraic formulae
$\varphi(x,b_{1}),\ldots,\varphi(x,b_{n})$ produce $n^{2}$ new elements
$c_{1},\ldots,c_{n^{2}}$ connected by $\\{a\\}$-orbits. Thus,
$c_{1},\ldots,c_{n^{2}}\in{\rm acl}_{n}({\rm acl}_{n}(\\{a\\}))$ whereas these
elements do not belong to ${\rm acl}_{n}(\\{a\\})$.

###### Remark 3.12

Clearly, Exchange property for ${\rm acl}_{n}(\cdot)$ can fail even if ${\rm
acl}(\cdot)$ satisfies it. Indeed, let $\Gamma$ be a bipartite graph with
parts $U$ and $V$ such that $|U|=n$, $|V|=m>n$ and each vertex in $U$ is
connected by an edge with each vertex in $V$. We have $a\in{\rm
acl}_{n}(\\{b\\})$ for any $a\in U$ and $b\in V$, whereas $b\in{\rm
acl}(\\{a\\})\setminus{\rm acl}_{n}(\\{a\\})$.

At the same time replacing ${\rm acl}_{n}(\cdot)$ by ${\rm acl}_{m}(\cdot)$ we
obtain that Exchange property. In fact, Exchange property for ${\rm
acl}_{m}(\cdot)$ is implied by Exchange property for the case ${\rm
acl}(\cdot)={\rm acl}_{m}(\cdot)$.

###### Remark 3.13

Let $T$ be a theory of an equivalence relation $E$. By the definition any
element $a$ of a finite $E$-class $X$ is $|X|$-algebraic over any singleton
$\\{b\\}\in X\setminus\\{a\\}$. Moreover, $a$ is $n$-algebraic over
$\emptyset$ if there are finitely many $E$-classes of the cardinality $|X|$.
Otherwise, if there are infinitely many $E$-classes of the cardinality $|X|$
these $E$-classes do not contain algebraic elements over $\emptyset$.

If the $E$-class $X$ is infinite it does not contain algebraic elements and
for each $A\subseteq X$, then ${\rm acl}(A)={\rm dcl}(A)=A$, and the
correspondent theory is quasi-Urbanik.

By Remark 3.13 we immediately obtain the following dichotomy for values ${\rm
deg}_{\rm acl}(T)$ of theories $T$ in the language $\\{E\\}$ of an equivalence
relation:

###### Proposition 3.14

For any theory $T$ of an equivalence relation $E$ and a set $A$ either ${\rm
deg}_{\rm acl}(A)\in\omega$, if $A$ consists of elements with bounded orbits
inside finite $E$-classes and between these $E$-classes, or ${\rm deg}_{\rm
acl}(A)=\infty$, otherwise.

###### Example 3.15

Let $\mathcal{M}$ be a structure in the language $\\{E\\}$ of equivalence
relation and consists of $E$-classes with unbounded distinct finite
cardinalities then the sets ${\rm acl}_{n}(\emptyset)$, $n\geq 1$, form an
unboundedly increasing sequence of $n$-algebraically closed sets. We observe
the same effect for ${\rm acl}_{n}(\emptyset)$, $n\geq 1$, where $A$ consists
of boundedly many elements in each $E$-class producing ${\rm deg}_{\rm
acl}(A)=\infty$ and ${\rm deg}_{\rm acl}({\rm Th}(\mathcal{M}))=\infty$ as
well.

The following example illustrates the possibility ${\rm deg}_{\rm
acl}(T)=\omega$.

###### Example 3.16

Let $\mathcal{M}$ be a structure in the language $\\{E\\}$ and consisting of
an equivalence relation $E$ with infinitely many $n$-element $E$-classes for
each $n\in\omega\setminus\\{0\\}$. By the definition all finite orbits over a
set $A\subseteq M$ are exhausted by automorphisms transforming elements in
$E$-classes $X\subseteq M$ with $X\cap A\neq\emptyset$. By the arguments in
Example 3.15 we have ${\rm deg}_{\rm acl}(T)=\infty$ for $T={\rm
Th}(\mathcal{M})$.

Now we expand the structure $\mathcal{M}$ till a structure
$\mathcal{M}^{\prime}$ by countably many ternary predicates $R$ in the
following way:

1) if $\models R(a,b,c)$ then $a$ and $b$ belong to disjoint $E$-classes
$E(a)$ and $E(b)$ and $c\in E(a)\cup E(b)$ is the unique solution of
$R(a,b,z)$, moreover, we require that each $c^{\prime}\in E(a)\cup E(b)$ has a
ternary symbol $R^{\prime}$ with $\models
R^{\prime}(a,b,c^{\prime})\wedge\exists^{=1}zR^{\prime}(a,b,z)$;

2) $\mathcal{M}^{\prime}$ has same orbits as $\mathcal{M}$ over subsets of
$E$-classes and one-element orbits only in $E(a)\cup E(b)$, for $E(a)\cap
E(b)=\emptyset$, over sets $A\subseteq E(a)\cup E(b)$ which have both elements
in $E(a)$ and in $E(b)$.

The structure $\mathcal{M}^{\prime}$ can be formed step-by-step using an
appropriate generic construction [13, 14].

By the definition for each $A\subseteq M^{\prime}$, ${\rm deg}_{\rm acl}(A)$
is finite, moreover, ${\rm deg}_{\rm acl}(A)=1$ for $A$ laying in several
finite $E$-classes. At the same time the values ${\rm deg}_{\rm acl}(A)$ are
unbounded in $\omega$ using subsets of finite $E$-classes. Therefore, ${\rm
deg}_{\rm acl}({\rm Th}(\mathcal{M}^{\prime}))=\omega$.

###### Theorem 3.17

$1.$ For any consistent theory $T$, ${\rm deg}_{\rm
acl}(T)\in((\omega+1)\setminus\\{0\\})\cup\\{\infty\\}$.

$2.$ For any $\lambda\in((\omega+1)\setminus\\{0\\})\cup\\{\infty\\}$ there is
a theory $T_{\lambda}$ such that ${\rm deg}_{\rm acl}(T_{\lambda})=\lambda$.

Proof. Item 1 holds since ${\rm deg}_{\rm acl}(T)$ is a supremum of values in
$(\omega\setminus\\{0\\})\cup\\{\infty\\}$.

2\. At first we consider theories $T$ of equivalence relations $E$. The values
${\rm deg}_{\rm acl}(T_{n})=n\in(\omega\setminus\\{0\\})$ are realized by $E$
on $n$-element sets $M$ with unique equivalence classes. Here each set
$A\subseteq M$ has ${\rm acl}(A)={\rm acl}_{n}(A)=M$. We obtain a similar
effect adding infinite $E$-classes, or taking finitely many finite $E$-classes
having a total of $n$ elements, or adding both these $E$-classes.

The value ${\rm deg}_{\rm acl}(T_{\infty})=\infty$ is confirmed by an
equivalence relation $E$ with infinitely many finite $E$-classes with
unbounded finite cardinalities. Here we take a set $A$ containing elements in
all these classes producing ${\rm acl}(A)\neq{\rm acl}_{n}(A)$ for any
$n\in\omega$.

The value ${\rm deg}_{\rm acl}(T_{\omega})=\omega$ is witnessed by Example
3.16.

###### Example 3.18

By the definition any definable element is $n$-algebraic, for any $n\geq 1$,
but not vice versa. For instance, taking a graph
$\Gamma=\langle\\{a_{1},a_{2}\\};\\{e\\}\rangle$, where $e$ is the edge
$[a_{1},a_{2}]$, we obtain ${\rm acl}(\emptyset)=\\{a_{1},a_{2}\\}$ witnessed
by the formula $x\approx x$, i.e., ${\rm deg}_{\rm acl}(\emptyset)={\rm
deg}_{\rm acl}({\rm Th}(\Gamma))=2$, whereas ${\rm dcl}(\emptyset)=\emptyset$,
since $a_{1}$ and $a_{2}$ are connected by an automorphism. It is a minimal
example, by inclusion, in a graph language.

We observe the same effect for a structure with at least two elements in the
empty language, or in the graph language with the empty relation, or in the
language of one unary function consisting of loops or of a cycle.

###### Example 3.19

For any linearly ordered structure $\mathcal{M}$ and a set $A\subseteq M$,

${\rm acl}(A)={\rm dcl}(A),$ (2)

since for any formula $\varphi(x,\overline{a})$ with finitely many solutions
the finite set $B$ of these solutions is linearly ordered and each element of
$B$ is defined by its position in that finite linear order. Thus, for the
linearly ordered structure $\mathcal{M}$, ${\rm deg}_{\rm
acl}(\mathcal{M})=1$, and it has a quasi-Urbanik theory.

Example 2.12 in [15] illustrates that for the circularly ordered structure
$c(\omega+\omega^{\ast}+{\bf Q}+\omega+\omega^{\ast}+{\bf Q})$, ${\rm
acl}(\emptyset)$ is a proper superset of ${\rm dcl}(\emptyset)$ contrasting
with the linearly ordered case, since the first elements of the copies of
$\omega$ are connected by an automorphism, and the last elements of the copies
of $\omega^{\ast}$ are connected by an automorphism, too.

At the same time, for any circularly ordered structure $\mathcal{M}$ and a
nonempty set $A\subseteq M$, ${\rm acl}(A)={\rm dcl}(A)$, since any element
$a\in A$ defines a linear order on $M\setminus\\{a\\}$ satisfying the equality
(2). Thus, for the circularly ordered structure $\mathcal{M}$, ${\rm deg}_{\rm
acl}({\rm Th}(\mathcal{M}))={\rm deg}_{\rm acl}(\emptyset)$. Adding copies of
$\omega+\omega^{\ast}+{\bf Q}$ to the circularly ordered structure
$c(\omega+\omega^{\ast}+{\bf Q}+\omega+\omega^{\ast}+{\bf Q})$ we can obtain
an arbitrary natural value ${\rm deg}_{\rm acl}(\emptyset)$ for a circularly
ordered theory.

Continuing the process with infinitely and densely many copies of
$\omega+\omega^{\ast}+{\bf Q}$ and marking some of them circularly by unary
predicates we can obtain a circularly ordered theory $T$ with

${\rm deg}_{\rm acl}(T)={\rm deg}_{\rm acl}(\emptyset)=\infty.$ (3)

We can, for instance, choose a prime number $p$ and mark $p$ copies
$C_{0},\ldots,C_{p-1}$ by a unary predicate $P_{0}$, then choose $p$ copies
$C_{i,0},\ldots,C_{i,p-1}$ between $C_{i}$ and $C_{(i+1)({\rm mod}p)}$, for
each $i\leq p-1$, and mark these copies by a unary predicate $P_{1}$.
Continuing the process with countably many disjoint unary predicates $P_{n}$,
$n\in\omega$, we obtain increasing finite cardinalities for orbits of the
first elements of the marked copies of $\omega$ witnessing the equalities (3).

We observe the similar effect for any $n$-spherically ordered structure
$\mathcal{M}$. In particular, ${\rm acl}(A)={\rm dcl}(A)$ for any set
$A\subseteq M$ of a cardinality $\geq n-2$, since in such a case a
$n$-spherical order is reduced to a linear one [16].

Definition. If for a theory $T$, ${\rm dcl}(A)={\rm acl}(A)$ for any set $A$
with $|A|\geq n$ then minimal such $n$ is called the ${\rm acl}$-${\rm
dcl}$-difference and denoted by ${\rm acl}$-${\rm dcl}_{\rm dif}(T)$. If such
natural $n$ does not exists, i.e., for any $n\in\omega$ there exists a set $A$
with $|A|\geq n$ and ${\rm acl}(A)\supset{\rm dcl}(A)$ then we put ${\rm
acl}$-${\rm dcl}_{\rm dif}(T)=\infty$.

###### Remark 3.20

By the definition and Example 3.19 we observe that ${\rm acl}$-${\rm dcl}_{\rm
dif}(T_{0})=0$ for any linearly ordered theory $T_{0}$, ${\rm acl}$-${\rm
dcl}_{\rm dif}(T)\leq 1$ for any circularly ordered theory $T$, with ${\rm
acl}$-${\rm dcl}_{\rm dif}(T_{1})=1$ if $T_{1}$ is circularly ordered with
unique $1$-type and a model with at least two elements. More generally, ${\rm
acl}$-${\rm dcl}_{\rm dif}(T)\leq n-2$ for any $n$-spherically ordered theory
$T$, where $n\geq 3$, with ${\rm acl}$-${\rm dcl}_{\rm dif}(T_{n-2})=n-2$ if,
for instance, $T_{n-2}$ is dense $n$-spherically ordered [17].

Taking a disjoint union [13, 12] of models $\mathcal{M}_{n-2}$ of
$n$-spherically ordered theories $T_{n-2}$, with unboundedly many $n$, we
obtain a structure $\mathcal{M}$ with ${\rm acl}$-${\rm dcl}_{\rm dif}({\rm
Th}(\mathcal{M}))=\infty$.

In view of Remark 3.20 we have the following:

###### Theorem 3.21

For any $\lambda\in\omega\cup\\{\infty\\}$ there is a theory $T_{\lambda}$
with ${\rm acl}$-${\rm dcl}_{\rm dif}(T_{\lambda})=\lambda$.

##  4 Algebraic sets and their degrees

Definition. Let $\mathcal{M}$ be a $L$-structure, $A,B\subseteq M$. The set
$B$ is called algebraic over $A$, or $A$-algebraic if it is the finite set of
solutions in $\mathcal{M}$ of a $L(A)$-formula $\varphi(x,\overline{a})$,
where $\varphi(x,\overline{y})$ is a $L$-formula and $\overline{a}\in A$,
i.e., $B$ consists of algebraic elements over $A$ witnessed by a fixed formula
$\varphi(x,\overline{a})$. If $A$ is empty then an $A$-algebraic set is called
algebraic.

The set $B$ is called $(A,u)$-algebraic if it is a union of $A$-algebraic
sets.

If $B$ is a union of $A$-algebraic subsets of cardinalities at most $n$, for
$n\in\omega$, then $B$ is called $(A,u,n)$-algebraic. If $B$ is both
$A$-algebraic and $(A,u,n)$-algebraic then $B$ is called $(A,n)$-algebraic.

The least $n$ for the $(A,u,n)$-algebraicity of $B$ is called the degree of
the $(A,u,n)$-algebraicity of $B$ and it is denoted by ${\rm deg}_{{\rm
alg},u}(B/A)$. If $B$ is $(A,n)$-algebraic then the value ${\rm deg}_{{\rm
alg},u}(B/A)$ is defined, too, with ${\rm deg}_{{\rm alg},u}(B/A)\leq n$.

At the same time an $(A,u)$-algebraic set $B$ admits the value ${\rm
deg}_{{\rm alg},u}(B/A)=\infty$, if $B$ is not represented as a union of
$A$-algebraic sets in bounded finite cardinalities.

The value ${\rm sup}\\{{\rm deg}_{{\rm alg},u}(B/A)\mid B$ is
$(A,u)$-algebraic$\\}$ is called the degree of the $(A,u,n)$-algebraicity and
denoted by ${\rm DEG}_{{\rm alg},u}(A)$.

We denote by $\mathcal{A}_{A,\mathcal{M}}$ the set of all $A$-algebraic sets
in $\mathcal{M}$, by $\mathcal{A}^{u}_{A,\mathcal{M}}$ the set of all
$(A,u)$-algebraic sets in $\mathcal{M}$, by $\mathcal{A}^{n}_{A,\mathcal{M}}$
the set of all $(A,n)$-algebraic sets in $\mathcal{M}$, and by
$\mathcal{A}^{u,n}_{A,\mathcal{M}}$ the set of all $(A,u,n)$-algebraic sets in
$\mathcal{M}$.

We omit $A$ above if it is empty.

Notice that the sets $\mathcal{A}_{A,\mathcal{M}}$,
$\mathcal{A}^{u}_{A,\mathcal{M}}$, $\mathcal{A}^{n}_{A,\mathcal{M}}$,
$\mathcal{A}^{u,n}_{A,\mathcal{M}}$ are preserved under elementary extensions
of $\mathcal{M}$. Therefore we omit the indexes $\mathcal{M}$ above, denoting
the sets above by $\mathcal{A}_{A}$, $\mathcal{A}^{u}_{A}$,
$\mathcal{A}^{n}_{A}$, $\mathcal{A}^{u,n}_{A}$.

The following assertion collects some properties of algebraic sets.

###### Proposition 4.1

$1.$ For any set $A$ and $m\leq n<\omega$,
$\mathcal{A}^{m}_{A}\subseteq\mathcal{A}^{n}_{A}\cap\mathcal{A}^{u,m}_{A}\subseteq\mathcal{A}_{A}\subseteq\mathcal{A}^{u}_{A}$,
$\mathcal{A}^{u,m}_{A}\subseteq\mathcal{A}^{n}_{A}\cap\mathcal{A}^{u,n}_{A}\subseteq\mathcal{A}^{u}_{A}$.

$2.$ If ${\rm deg}_{{\rm alg},u}(B/A)$ is defined, i.e.
$B\in\mathcal{A}^{u}_{A}$, then $B\subseteq{\rm acl}(A)$. The converse holds
if $B$ consists of some finite orbits (in a saturated extension of
$\mathcal{M}$) of elements under $A$-automorphisms.

$3.$ (Monotony) For any $B_{1},B_{2}\in\mathcal{A}^{u}_{A}$ with
$B_{1}\subseteq B_{2}$, ${\rm deg}_{{\rm alg},u}(B_{1}/A)\leq{\rm deg}_{{\rm
alg},u}(B_{2}/A)$.

$4.$ (Monotony) If $B\in\mathcal{A}^{u}_{A_{1}}$ and $A_{1}\subseteq A_{2}$
then ${\rm deg}_{{\rm alg},u}(B/A_{1})\geq{\rm deg}_{{\rm alg},u}(B/A_{2})$.

$5.$ For any set $A$, ${\rm DEG}_{{\rm alg},u}(A)={\rm deg}_{{\rm alg},u}({\rm
acl}(A)/A)$.

$6.$ For any sets $A$ and $B$, ${\rm deg}_{\rm alg}(B/A)=0$ iff $B=\emptyset$.

$7.$ For any sets $A$ and $B$, ${\rm deg}_{\rm alg}(B/A)=1$ iff $\emptyset\neq
B\subseteq{\rm dcl}(A)$.

$8.$ If $A$ is algebraically closed then ${\rm DEG}_{{\rm alg},u}(A)=0$ if $A$
is empty, and ${\rm DEG}_{{\rm alg},u}(A)=1$ if $A$ is nonempty.

Proof. Item 1 holds since any $m$-algebraic element over $A$ is algebraic over
$A$, and, moreover, $n$-algebraic over $A$, for any $m\leq n$.

Item 2 is satisfied as elements of $\mathcal{A}^{u}_{A}$ consist of some
finite orbits of elements under $A$-automorphisms.

The monotonies are true by the definition.

Item 5 is implied by Item 2 and Monotony (Item 3).

Item 6 is obvious.

Item 7 holds by Item 6 and the property both for ${\rm deg}_{\rm alg}(B/A)=1$
and for $B\subseteq{\rm dcl}(A)$ that $B$ is composed as a union of
$A$-definable singletons.

Item 8 is implies by $A={\rm dcl}(A)$ and Items 6 and 7.

###### Corollary 4.2

If $\mathcal{M}$ is an algebra, then for any $A\subset M$ the universe $M(A)$
of the subalgebra $\mathcal{M}(A)$ of $\mathcal{M}$ generated by $A$ is
$(A,u)$-algebraic with ${\rm deg}_{{\rm alg},u}(M(A)/A)=1$.

Proof. Any element $a$ of $M(A)$ is represented by a term
$t(a_{1},\ldots,a_{n})$, where $a_{1},\ldots,a_{n}\in A$. Therefore
$M(A)\subseteq{\rm dcl}(A)$. Thus $M(A)$ is represented as a union of
singletons $\\{a\\}\in\mathcal{A}^{1}_{A}\subseteq\mathcal{A}^{u}_{A}$ whence
$M(A)\in\mathcal{A}^{u}_{A}$ and ${\rm deg}_{{\rm alg},u}(M(A)/A)=1$ in view
of Item 7 in Proposition 4.1.

###### Remark 4.3

For any set $A$, ${\rm DEG}_{{\rm alg},u}(A)\in\omega\cup\\{\infty\\}$, since
any $(A,u)$-algebraic set is either represented as a union of
$(A,u,n)$-algebraic sets for some $n\in\omega$, or such $n$ does not exists
for some $(A,u)$-algebraic set $B$. In the first case we have ${\rm DEG}_{{\rm
alg},u}(A)\in\omega$, and in the second one ${\rm DEG}_{{\rm
alg},u}(A)=\infty$.

For an illustration we show that all values ${\rm DEG}_{{\rm
alg},u}(A)\in\omega\cup\\{\infty\\}$ are realized for appropriate sets $A$.

Indeed, taking an equivalent relation $E$ with infinitely many equivalence
classes of cardinality $n+1$, we obtain ${\rm DEG}_{{\rm alg},u}(\\{a\\})=n$
for any singleton $A=\\{a\\}$, where there are exactly four $A$-algebraic
sets: $\emptyset$, $\\{a\\}$, $E(a)\setminus\\{a\\}$, $E(a)$.

Now let $E_{n}$, $n\in\omega$, be an ascending chain of equivalence relations
such that $E_{0}$-classes are singletons and each $E_{n+1}$-class consists of
exactly two $E_{n}$-classes. Taking an arbitrary singleton $A=\\{a\\}$ we
obtain ${\rm DEG}_{{\rm alg},u}(\\{a\\})=\infty$, witnessed by the set
$B=\bigcup\limits_{n\in\omega}E_{n}(a)$, since all $E_{n}$-classes are finite
and have unboundedly increasing cardinalities.

We obtain a similar effect taking disjoint unary predicates $P_{n}$ with
$|P_{n}|=n$, $n\in\omega$. Here ${\rm DEG}_{{\rm alg},u}(\emptyset)=\infty$,
and ${\rm DEG}_{{\rm alg},u}(\emptyset)\in\omega$, if the language is
restricted to finitely many predicate symbols.

##  5 Further variations of algebraic and definable closures and their
connections

Definition. 1\. Let $\Delta$ be a set of formulae of a theory $T$. For a
models $\mathcal{M}\models T$ and a set $A\subseteq M$ the union of sets of
solutions of formulae $\varphi(x,\overline{a})$, where
$\varphi(x,\overline{y})\in\Delta$ and $\overline{a}\in A$, such that
$\models\exists^{=n}x\>\varphi(x,\overline{a})$ for some $n\in\omega$
(respectively, $\models$ $\exists^{=1}x\>\varphi(x,\overline{a})$) is said to
be a _$\Delta$ -algebraic ($\Delta$-definable or $\Delta$-definitional)
closure_ of $A$. The $\Delta$-algebraic closure of $A$ is denoted by ${\rm
acl}^{\Delta}(A)$ and its $\Delta$-definable ($\Delta$-definitional) closure,
by ${\rm dcl}^{\Delta}(A)$.

In such a case we say that the formulae $\varphi(x,\overline{a})$ witness that
$\Delta$-algebraic / $\Delta$-definable ($\Delta$-definitional) closure, and
these formulae are called $\Delta$-algebraic / $\Delta$-defining.

Any element $b\in{\rm acl}^{\Delta}(A)$ (respectively, $b\in{\rm
dcl}^{\Delta}(A)$) is called $\Delta$-algebraic ($\Delta$-definable or
$\Delta$-definitional) over $A$. If the set $A$ is fixed or empty, we just say
that $b$ is $\Delta$-algebraic, $\Delta$-definable, or $\Delta$-definitional.

2\. If ${\rm dcl}^{\Delta}(A)={\rm acl}^{\Delta}(A)$, ${\rm
cl}_{1}^{\Delta}(A)$ denotes their common value. In such a case we say that
$A$ is $\Delta$-quasi-Urbanik.

3\. If $A={\rm acl}^{\Delta}(A)$ (respectively, $A={\rm dcl}^{\Delta}(A)$)
then $A$ is called $\Delta$-algebraically ($\Delta$-definably) closed.

We combine $\Delta$-algebraic closures and $n$-algebraic closure as follows.

Definition. 1\. For $n\in\omega\setminus\\{0\\}$, a set $A\subseteq M$ and a
set $\Delta$ of formulae an element $b$ is called $(\Delta,n)$-algebraic over
$A$, if $b\in{\rm acl}^{\Delta}(A)$ and it is witnessed by a formula
$\varphi(x,\overline{a})$, for some $\overline{a}\in A$ and
$\varphi(x,\overline{y})\in\Delta$, with at most $n$ solutions.

2\. The set of all $(\Delta,n)$-algebraic elements over $A$ is denoted by
${\rm acl}^{\Delta}_{n}(A)$.

3\. If $A={\rm acl}^{\Delta}_{n}(A)$ then $A$ is called
$(\Delta,n)$-algebraically closed.

4\. If ${\rm acl}^{\Delta}(A)={\rm acl}^{\Delta}_{n}(A)$ then minimal such $n$
is called the $\Delta$-degree of algebraization over the set $A$ and it is
denoted by ${\rm deg}^{\Delta}_{\rm acl}(A)$. If that $n$ does not exist then
we put ${\rm deg}^{\Delta}_{\rm acl}(A)=\infty$. The supremum of values ${\rm
deg}^{\Delta}_{\rm acl}(A)$ with respect to all sets $A$ of given theory $T$
is denoted by ${\rm deg}^{\Delta}_{\rm acl}(T)$ and called the $\Delta$-degree
of algebraization of the theory $T$.

5\. A theory $T$ with ${\rm deg}^{\Delta}_{\rm acl}(T)=1$, i.e., with quasi-
Urbanik sets $A$ of $T$ only, is called $\Delta$-quasi-Urbanik, and the models
$\mathcal{M}$ of $T$ are $\Delta$-quasi-Urbanik, too.

The following remark collects some obvious properties related to the operators
${\rm acl}^{\Delta}$ and ${\rm acl}^{\Delta}_{n}$.

###### Remark 5.1

$1.$ (Monotony) If $m\leq n$, $\Delta_{1}\subseteq\Delta_{2}$, and
$A_{1}\subseteq A_{2}$ then ${\rm acl}^{\Delta_{1}}_{m}(A_{1})\subseteq{\rm
acl}^{\Delta_{2}}_{n}(A_{2})$.

$2.$ ($\Delta$-reduction) For any set $A$, ${\rm acl}(A)={\rm
acl}^{\Delta}(A)$ (respectively, ${\rm acl}_{n}(A)={\rm acl}_{n}^{\Delta}(A)$,
where $n\in\omega\setminus\\{0\\}$) iff for any $a\in{\rm acl}(A)$ ($a\in{\rm
acl}_{n}(A)$) it is witnessed by a formula in $\Delta$.

Definition [21]). A theory $T$ is said to be _$\Delta$ -based_, where $\Delta$
is some set of formulas without parameters, if any formula of $T$ is
equivalent in $T$ to a Boolean combination of formulae of $\Delta$.

For $\Delta$-based theories $T$, it is also said that $T$ has quantifier
elimination or quantifier reduction up to $\Delta$.

Let $\Delta$ be a set of formulae of a theory $T$.

###### Remark 5.2

1\. For the theory $T$, the operators ${\rm acl}^{\Delta}$ and ${\rm
acl}^{\Delta}_{n}$ coincide iff finite definable sets $A$ for formulae
$\varphi(x,\overline{y})\in\Delta$ with $A=\varphi(\mathcal{M},\overline{a})$,
$\mathcal{M}\models T$ are covered by definable sets $A_{i}$ for formulae
$\varphi_{i}(x,\overline{y})\in\Delta$ with
$A_{i}=\varphi_{i}(\mathcal{M},\overline{a})$ having at most $n$ elements.

2\. If a theory $T$ is $\Delta$-based then both ${\rm
acl}^{\Delta^{\prime}}={\rm acl}$ and ${\rm acl}^{\Delta^{\prime}}_{n}={\rm
acl}_{n}$, where

$\Delta^{\prime}=\left\\{\bigwedge\limits_{i}\varphi_{i}^{\delta_{i}}\mid\varphi_{i}\in\Delta,\delta_{i}\in\\{0,1\\}\right\\}.$

###### Proposition 5.3

$1.$ The inclusion $A\subseteq{\rm acl}^{\Delta}(A)$ (respectively,
$A\subseteq{\rm acl}^{\Delta}_{n}(A)$) holds for any set $A$ if and only if
$\Delta$ is ${\rm acl}$-reflexive ($({\rm acl},n)$-reflexive), i.e., for any
element $a$ there is $\varphi(x,y)\in\Delta$ with
$\models\varphi(a,a)\wedge\exists^{\leq m}x\,\varphi(x,a)$ for some
$m\in\omega$ (for some $m\leq n$, where $n\in\omega\setminus\\{0\\}$).

$2.$ The inclusion ${\rm acl}^{\Delta}({\rm acl}^{\Delta}(A))\subseteq{\rm
acl}^{\Delta}(A)$ (respectively, ${\rm acl}^{\Delta}_{n}({\rm
acl}^{\Delta}_{n}(A))\subseteq{\rm acl}^{\Delta}_{n}(A)$) holds for any set
$A$ if and only if $\Delta$ is ${\rm acl}$-transitive ($({\rm
acl},n)$-transitive), i.e., for any tuples $\overline{a}\in A$,
$\overline{b}\in{\rm acl}^{\Delta}(A)$ ($\overline{b}\in{\rm
acl}^{\Delta}_{n}(A)$) and a formula $\varphi(x,\overline{z})\in\Delta$ with
$\models\varphi(c,\overline{b})\wedge\exists^{<\omega}x\varphi(x,\overline{b})$
$(\models\varphi(c,\overline{b})\wedge\exists^{\leq
n}x\varphi(x,\overline{b}))$ there is a formula
$\psi(x,\overline{y})\in\Delta$ with
$\models\psi(c,\overline{a})\wedge\exists^{<\omega}x\psi(x,\overline{a})$
$(\models\psi(c,\overline{a}\wedge\exists^{\leq n}x\psi(x,\overline{a})$,
where $n\in\omega\setminus\\{0\\})$.

$3.$ The operators ${\rm acl}^{\Delta}$ and ${\rm acl}^{\Delta}_{n}$ satisfy
Finite character.

$4.$ The operator ${\rm acl}$ (${\rm acl}_{n}$) has Exchange property iff any
operator ${\rm acl}_{n}^{\Delta}$ (${\rm acl}_{n}^{\Delta}$) is extensible
till some ${\rm acl}^{\Delta^{\prime}}$ (${\rm acl}^{\Delta^{\prime}}_{n}$)
with $\Delta\subseteq\Delta^{\prime}$ and Exchange property. Here either ${\rm
acl}^{\Delta^{\prime}}$ (${\rm acl}^{\Delta^{\prime}}_{n}$ with given $n$) has
a proper extension or does not have proper extensions depending on the given
theory.

Proof is obvious using the definition of operators ${\rm acl}^{\Delta}$ and
${\rm acl}^{\Delta}_{n}$.

Proposition 5.3 characterizes basic properties of closure operators. If the
operator ${\rm acl}^{\Delta}$ (respectively, ${\rm acl}^{\Delta}_{n}$)
satisfies the conditions described in Items 1 and 2 of Proposition 5.3 then it
is called regular. If, additionally, the condition in Item 4 holds, then that
operator is called pregeometric.

Proposition 5.3 immediately implies:

###### Corollary 5.4

The operator ${\rm acl}^{\Delta}$ (respectively, ${\rm acl}^{\Delta}_{n}$)
defines a pregeometry on the universe of given structure iff it is
pregeometric, i.e., it is regular and satisfies Exchange property.

###### Remark 5.5

Any operator ${\rm acl}^{\Delta}$ is extensible till transitive one ${\rm
acl}^{\Delta^{\prime}}$ by the transitive closure forming $\Delta^{\prime}$ by
adding to $\Delta$ all formulae

$\exists
x_{1},\ldots,x_{n}\left(\psi(x,x_{1},\ldots,x_{n})\wedge\bigwedge\limits_{i=1}^{n}\varphi_{i}(x_{i},\overline{y})\right)$
(4)

as in (1), where $\varphi_{i}$ and $\psi$ either belong to $\Delta$ or were
added before, i.e., $\Delta^{\prime}$ is the set of formulae obtained from
$\Delta$ in the calculus with the rules:

$\displaystyle\frac{\varphi_{1},\ldots,\varphi_{n},\psi}{\exists
x_{1},\ldots,x_{n}\left(\psi(x,x_{1},\ldots,x_{n})\wedge\bigwedge\limits_{i=1}^{n}\varphi_{i}(x_{i},\overline{y})\right)},\mbox{\
\ for }n\in\omega.$

Using Monotony in Remark 5.1 we have ${\rm acl}^{\Delta}(A)\subseteq{\rm
acl}^{\Delta^{\prime}}(A)$ for any set $A$ in the given structure. Taking
$\Delta^{\prime\prime}=\Delta^{\prime}\cup\\{x\approx y\\}$ we obtain a
regular operator ${\rm acl}^{\Delta^{\prime\prime}}$.

We have a similar regular operator ${\rm acl}^{\Delta^{\prime\prime}}_{1}$
starting with the operator ${\rm acl}^{\Delta}_{1}$.

In contrast with ${\rm acl}^{\Delta}$ and ${\rm acl}^{\Delta}_{1}$ the
transitivity of ${\rm acl}^{\Delta^{\prime\prime}}_{n}$, for $n\geq 2$, can
fail under the transitive closure above, since compositions of $n$-algebraic
formulae may not be $n$-algebraic.

In view of properties above a hierarchy of regular and pregeometric algebraic
closure operators ${\rm acl}^{\Delta}$, ${\rm acl}^{\Delta}_{n}$ arises
depending on chosen sets $\Delta$ of formulae and bounds $n$ of cardinalities
for sets of solutions of algebraic formulae in $\Delta$. This hierarchy
includes various degrees ${\rm acl}$-${\rm dcl}_{\rm dif}(T)$ of
algebraization.

Definition. Two sets $\Delta_{1}$ and $\Delta_{2}$ of formulae are called
${\rm acl}$-equivalent (respectively, ${\rm acl}_{n}$-equivalent), denoted by
$\Delta_{1}\sim\Delta_{2}$ ($\Delta_{1}\sim_{n}\Delta_{2}$), if ${\rm
acl}^{\Delta_{1}}={\rm acl}^{\Delta_{2}}$ (${\rm acl}^{\Delta_{1}}_{n}={\rm
acl}^{\Delta_{2}}_{n}$).

###### Remark 5.6

Equivalent sets preserve the regularity and pregeometricity. Moreover,
equivalence classes $\sim\\!\\!(\Delta_{1})$ and $\sim\\!\\!(\Delta_{2})$
(respectively, $\sim_{n}\\!\\!(\Delta_{1})$ and $\sim_{n}\\!\\!(\Delta_{2})$)
allow to choose coordinated representatives
$\Delta^{\prime}_{1}\in\,\,\sim\\!\\!(\Delta_{1})$ and
$\Delta^{\prime}_{2}\in\,\,\sim\\!\\!(\Delta_{2})$ (respectively,
$\Delta^{\prime}_{1}\in\,\,\sim_{n}\\!\\!(\Delta_{1})$ and
$\Delta^{\prime}_{2}\in\,\,\sim_{n}\\!\\!(\Delta_{2})$) such that for regular
${\rm acl}^{\Delta^{\prime}_{1}}$ and ${\rm acl}^{\Delta^{\prime}_{2}}$, ${\rm
acl}^{\Delta^{\prime}_{1}}\cap\,\,{\rm acl}^{\Delta^{\prime}_{2}}$ (${\rm
acl}_{n}^{\Delta^{\prime}_{1}}\cap\,\,{\rm acl}_{n}^{\Delta^{\prime}_{2}}$) is
regular, too, and ${\rm acl}^{\Delta^{\prime}_{1}}\cap\,\,{\rm
acl}^{\Delta^{\prime}_{2}}={\rm
acl}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$ (${\rm
acl}_{n}^{\Delta^{\prime}_{1}}\cap{\rm acl}_{n}^{\Delta^{\prime}_{2}}={\rm
acl}_{n}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$). Indeed, it suffices
to add to $\Delta^{\prime}_{1}$ and to $\Delta^{\prime}_{2}$ same formulae
witnessing common finite definable sets with respect to ${\rm
acl}^{\Delta_{1}}$ and ${\rm acl}^{\Delta_{2}}$ (${\rm acl}_{n}^{\Delta_{1}}$
and ${\rm acl}_{n}^{\Delta_{2}}$).

Thus, the binary operations $\wedge$ and $\wedge_{n}$ of intersection arise
mapping the pairs $({\rm acl}^{\Delta_{1}},{\rm acl}^{\Delta_{2}})$
(respectively $({\rm acl}_{n}^{\Delta_{1}},{\rm acl}_{n}^{\Delta_{2}})$) to
${\rm acl}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$ (${\rm
acl}_{n}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$).

For a structure $\mathcal{M}$ we define derived structures $\mathcal{SL}_{\rm
acl}(\mathcal{M})$ and $\mathcal{SL}_{{\rm acl}_{n}}(\mathcal{M})$,
$n\in\omega$, in the following way. The universe ${\rm SL}_{\rm
acl}(\mathcal{M})$ (respectively, ${\rm SL}_{{\rm acl}_{n}}(\mathcal{M})$) of
$\mathcal{SL}_{\rm acl}(\mathcal{M})$ ($\mathcal{SL}_{{\rm
acl}_{n}}(\mathcal{M})$) consists of all regular operators ${\rm
acl}^{\Delta}$ and ${\rm acl}^{\Delta}_{n}$ for $\mathcal{M}$, and the
language consists of the symbol $\wedge$ ($\wedge_{n}$) for the operation of
intersection.

If $\mathcal{M}$ is sufficiently saturated then the operators ${\rm
acl}^{\Delta}$ and ${\rm acl}^{\Delta}_{n}$ admit syntactical descriptions by
families of formulae, and it induces structures $\mathcal{SL}_{\rm acl}(T)$
and $\mathcal{SL}_{{\rm acl}_{n}}(T)$, for the theory $T={\rm
Th}(\mathcal{M})$, which are isomorphic to $\mathcal{SL}_{\rm
acl}(\mathcal{M})$ and $\mathcal{SL}_{{\rm acl}_{n}}(\mathcal{M})$,
respectively.

###### Proposition 5.7

$1.$ Any structure $\mathcal{SL}_{\rm acl}(T)$ is a lower semilattice with the
least and the greatest elements.

$2.$ Any structure $\mathcal{SL}_{{\rm acl}_{n}}(T)$ is a lower semilattice
with the least element. It has the greatest element if ${\rm acl}_{n}$ is
regular.

Proof. By the definition both intersections ${\rm
acl}^{\Delta^{\prime}_{1}}\cap\,\,{\rm acl}^{\Delta^{\prime}_{2}}={\rm
acl}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$ and ${\rm
acl}_{n}^{\Delta^{\prime}_{1}}\cap{\rm acl}_{n}^{\Delta^{\prime}_{2}}={\rm
acl}_{n}^{\Delta^{\prime}_{1}\cap\Delta^{\prime}_{2}}$ in Remark 5.6 are
infimum for $({\rm acl}^{\Delta_{1}}$, ${\rm acl}^{\Delta_{2}})$ and $({\rm
acl}_{n}^{\Delta_{1}}$, ${\rm acl}_{n}^{\Delta_{2}})$, respectively. The
operator ${\rm acl}^{\\{x\approx y\\}}$ is the least element both for
$\mathcal{SL}_{\rm acl}(T)$ and $\mathcal{SL}_{{\rm acl}_{n}}(T)$. The
operator ${\rm acl}$ is the greatest element for $\mathcal{SL}_{\rm acl}(T)$.
And ${\rm acl}_{n}$ is the greatest element for $\mathcal{SL}_{\rm
acl_{n}}(T)$ if ${\rm acl}_{n}$ is regular.

Since ${\rm acl}_{1}$ is always regular, Proposition 5.7 implies the
following:

###### Corollary 5.8

Any structure $\mathcal{SL}_{{\rm acl}_{1}}(T)$ is a lower semilattice with
the least and the greatest elements.

The following example illustrates that $\mathcal{SL}_{{\rm acl}_{n}}(T)$ may
not have the greatest element.

###### Example 5.9

Let $\mathcal{H}=\langle M,Z,E_{1},E_{2}\rangle$, $Z\subseteq\mathcal{P}(M)$,
be a hypergraph [9] with colored hyperedges, being equivalence classes of
equivalence relations $E_{1}$ and $E_{2}$, and satisfying the following
conditions:

i) each hyperedge consists of three elements;

ii) any element belongs to exactly two hyperedges, and one of these hyperedges
is marked by $E_{1}$ and another one by $E_{2}$;

iii) the hypergraph does not have cycles.

For the structure $\mathcal{M}=\langle M,E_{1},E_{2}\rangle$ we have regular
$\subseteq$-incomparable operators ${\rm acl}^{\\{x\approx
y,E_{i}(x,y)\\}}_{2}$, $i=1,2$, which do not have proper regular extensions,
since the operator ${\rm acl}_{2}$ on $M$ is not transitive. Indeed, taking an
arbitrary element $a\in M$ we have a $2$-algebraic formulae
$E_{1}(x,a)\wedge\neg(x\approx a)$ and $E_{1}(x,a)\wedge\neg(x\approx a)$. But
their composition is not $2$-algebraic having $4$ solutions.

Thus the semilattice $\mathcal{SL}_{{\rm acl}_{2}}(\mathcal{M})$ consists of
three operators: the least one ${\rm acl}^{\\{x\approx y\\}}_{2}$ and two its
$\subseteq$-incomparable extensions ${\rm acl}^{\\{x\approx
y,E_{i}(x,y)\\}}_{2}$, $i=1,2$. Moreover, since cardinalities of isolating
algebraic formulae $\varphi(x,a)$ are unbounded, for iterated compositions of
$E_{1}(x,y)$ and $E_{2}(x,y)$, operators ${\rm acl}_{n}$, for $n\geq 3$, are
not regular, too, implying that the semilattices $\mathcal{SL}_{{\rm
acl}_{n}}(\mathcal{M})$ also consist of three operators and do not have the
greatest elements. Adding to $\mathcal{SL}_{{\rm acl}_{n}}(\mathcal{M})$ the
operator ${\rm acl}$ we obtain the $4$-element semilattice $\mathcal{SL}_{{\rm
acl}}(\mathcal{M})$ forming a Boolean algebra. Notice also that
$\mathcal{SL}_{{\rm acl}_{1}}(\mathcal{M})$ is a singleton consisting of the
operator ${\rm acl}^{\\{x\approx y\\}}_{1}$.

We obtain the similar effect increasing the cardinality of $E_{i}$-classes
till natural $m\geq 4$. At the same time, if $m=2$, then $\mathcal{SL}_{{\rm
acl}_{1}}(\mathcal{M})$ forms a $4$-element lattice with the least and the
greatest elements.

Thus for the theories $T_{m}$ of structures $\mathcal{M}_{m}=\langle
M_{m},E_{1},E_{2}\rangle$ consisting of $m$-element hyperedges marked by
$E_{1}$ and $E_{2}$ as above, either ${\rm deg}_{\rm acl}(T_{m})=1$, if $m=1$
or $m=2$, or ${\rm deg}_{\rm acl}(T_{m})=\infty$, if $m\geq 3$.

Now we define expansions of the structures $\mathcal{SL}_{\rm
acl}(\mathcal{M})$ and $\mathcal{SL}_{{\rm acl}_{n}}(\mathcal{M})$,
$n\in\omega$, by the operation of union $\vee$. For the regular operators
${\rm acl}^{\Delta_{1}}$ and ${\rm acl}^{\Delta_{2}}$ (respectively, ${\rm
acl}_{n}^{\Delta_{1}}$ and ${\rm acl}_{n}^{\Delta_{2}}$) we put ${\rm
acl}^{\Delta_{1}}\vee{\rm acl}^{\Delta_{2}}={\rm acl}^{\Delta}$ (${\rm
acl}_{n}^{\Delta_{1}}\vee{\rm acl}_{n}^{\Delta_{2}}={\rm acl}_{n}^{\Delta}$),
where $\Delta$ is the transitive closure of $\Delta_{1}\cup\Delta_{2}$. We
denote these expansions of $\mathcal{SL}_{\rm acl}(\mathcal{M})$ and
$\mathcal{SL}_{{\rm acl}_{n}}(\mathcal{M})$ by $\mathcal{L}_{\rm
acl}(\mathcal{M})$ and $\mathcal{L}_{{\rm acl}_{n}}(\mathcal{M})$,
respectively. Again choosing $\mathcal{M}$ sufficiently saturated we obtain
structures $\mathcal{L}_{\rm acl}(T)$ and $\mathcal{L}_{{\rm acl}_{n}}(T)$ for
$T={\rm Th}(\mathcal{M})$.

Since the union $\vee$ and the intersection $\wedge$ correspond to appropriate
union and intersection of $\Delta_{i}$, in view of Proposition 5.7 we have the
following:

###### Theorem 5.10

The structure $\mathcal{L}_{\rm acl}(T)$ is a distributive lattice with the
least and the greatest elements.

Notice that the structures $\mathcal{L}_{{\rm acl}_{n}}(\mathcal{M})$ can be a
distributive lattices, too, taking, for instance, finite $\mathcal{M}$ and
$n\geq|M|$. At the same time, as Example 5.9 shows, in general case the
structures $\mathcal{L}_{{\rm acl}_{n}}(\mathcal{M})$ may be not lattices.

Recall that for a lattice $\mathcal{L}$, the lattice height (width) is the
supremum of cardinalities for (anti)chains. We denote these characteristics by
$h(\mathcal{L})$ and $w(\mathcal{L})$, respectively,

The following theorem shows that height and width of $\mathcal{L}_{\rm
acl}(T)$ can be unbounded.

###### Theorem 5.11

For any cardinality $\lambda>0$ there is a lattice $\mathcal{L}_{\rm acl}(T)$
with $h(\mathcal{L}_{\rm acl}(T))=\lambda$ and $w(\mathcal{L}_{\rm
acl}(T))=\lambda$.

Proof. The case $\lambda=1$ is realized by an arbitrary theory $T$ of a
singleton. The case $\lambda=2$ is realized in Example 5.9. Now for
$\lambda\geq 3$ we extend a structure $\mathcal{N}\equiv\mathcal{M}=\langle
M,E_{1},E_{2}\rangle$ for Example 5.9 till a structure $\mathcal{N}^{\prime}$
with $\lambda$ equivalence relations $E_{i}$, $i<\lambda$, satisfying the
following conditions:

i) each $E_{i}$-class contains 3 elements;

ii) for every $E_{i}$-class $X$ and $E_{j}$-class $Y$, where $i\neq j$,
$|X\cap Y|\leq 1$;

iii) the hypergraph $\langle M,Z\rangle$, where $Z$ is the set of all
$E_{i}$-classes, $i<\lambda$.

For the theory $T={\rm Th}(\mathcal{N}^{\prime})$ the lattice
$\mathcal{L}_{\rm acl}(T)$ has $\lambda$ atoms ${\rm acl}^{\\{x\approx
y,E_{i}(x,y)\\}}$, $i<\lambda$, witnessing $w(\mathcal{L}_{\rm
acl}(T))=\lambda$. Collecting formulae $E_{i}(x,y)$ we obtain a chain of
$\lambda$ operators ${\rm acl}^{\Delta_{\mu}}$, where
$\Delta_{\mu}={\\{x\approx y\\}\cup\\{E_{i}\mid i<\mu\\}}$, $\mu\leq\lambda$,
witnessing that $h(\mathcal{L}_{\rm acl}(T))=\lambda$.

##  6 Conclusion

We studied variations of algebraic closure, properties and possibilities of
these variations, of degree of algebraization, and of difference between
definable and algebraic closures. These possibilities are illustrated by a
series of examples. Algebraic sets and their degrees are studied. Hierarchy of
operators of algebraic closures relative sets is considered, semilattices and
lattices for families of these operators are introduced and some
characteristics of these structures are described.

It would be natural to describe degrees of algebraization and difference
between definable and algebraic closures for natural classes of structures and
their theories. It would be also interesting to describe possibilities of
Hasse diagrams for semilattices and lattices for families of operators of
algebraic closure.

## References

  * [1] Shelah S. Classification theory and the number of non-isomorphic models. Amsterdam : North-Holland, 1990. — 705 p.
  * [2] Hodges W. Model theory. Cambridge : Cambridge University Press, 1993\. — 772 p.
  * [3] Pillay A. Geometric Stability Theory. Oxford : Clarendon Press, 1996\. — 361 p.
  * [4] Marker D. Model Theory: An Introduction. New York : Springer-Verlag, 2002. — Graduate texts in Mathematics. — Vol. 217. — 342 p.
  * [5] Tent K., Ziegler M. A Course in Model Theory. Cambridge, Cambridge University Press, 2012. — 248 p.
  * [6] Birkhoff G. Lattice Theory. 3rd edition. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, 1967.
  * [7] G. Grätzer, General Lattice Theory. Pure and Applied Mathematics, vol. 75, Academic Press, New York, 1978; Mathematische Reihe, Band 52, Birkhäuser Verlag, Basel; Akademie Verlag, Berlin, 1978.
  * [8] Burris S., Sankappanavar H.P. A Course in Universal Algebra. Dover Publications, Incorporated, 2012. — 288 p.
  * [9] Emelichev V.A., Melnikov O.I., Sarvanov V.I., Tyshkevich R.I. Lectures on Graph Theory. Moscow, Nauka, 1990. — 384 p. [in Russian]
  * [10] Handbook of Mathematical Logic / ed. J. Barwise. — Moscow : Nauka, 1982. — Vol. 1. Model Theory. — 392 p. [in Russian]
  * [11] Zil’ber B.I. Hereditarily transitive groups and quasi-Urbanik structures, Trudy Inst. Mat. Sib. Otd. AN SSSR, 8 (1988), 58–77.
  * [12] Woodrow R.E. Theories with a finite number of countable models and a small language. Ph. D. Thesis, Simon Fraser University, 1976. — 99 p.
  * [13] Sudoplatov S.V. Classification of Countable Models of Complete Theories. Novosibirsk, NSTU, 2018.
  * [14] Sudoplatov S.V. Syntactic approach to constructions of generic models, Algebra and Logic, 46:4 (2007), 134–146.
  * [15] Kulpeshov B.Sh., Macpherson H.D. Minimality conditions on circularly ordered structures, Mathematical Logic Quarterly, 51:4 (2005), 377–399.
  * [16] Sudoplatov S.V. Minimality conditions, topologies, and ranks for spherically ordered theories, Siberian Electronic Mathematical Reports, 20:2 (2023), 600–615.
  * [17] Kulpeshov B.Sh., Sudoplatov S.V. Spherical orders, properties and countable spectra of their theories, Siberian Electronic Mathematical Reports, 20:2 (2023), 588–599.
  * [18] Ershov Yu.L., Palyutin E.A. Mathematical logic. — Moscow : FIZMATLIT, 2011. — 356 p. [in Russian]
  * [19] Sudoplatov S.V. Group Polygonometries. Novosibirsk, NSTU, 2013.
  * [20] Sudoplatov S.V. Models of cubic theories, Bulletin of the Section of Logic, 43:1–2 (2014), 19–34.
  * [21] Palyutin E.A., Saffe J., Starchenko S.S. Models of superstable Horn theories, Algebra and Logic, 24:3 (1985), 171–210.

Sergey V. Sudoplatov

Sobolev Institute of Mathematics,

Novosibirsk State Technical University,

Novosibirsk, Russia

E-mail<EMAIL_ADDRESS>