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A class of algebraic systems of the same type. All systems of a given type are assumed to be written in a given signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116701.png" /> and are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116702.png" />-systems. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116703.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116704.png" />-systems is called abstract if, whenever it contains a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116705.png" />, it also contains all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116706.png" />-systems isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116707.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116708.png" /> be an abstract class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a0116709.png" />-systems. One says that an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167010.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167011.png" /> has a local set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167012.png" />-subsystems if there exists an inclusion-directed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167013.png" /> of subsystems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167014.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167015.png" /> which cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167016.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167017.png" />) and which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167018.png" />. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167019.png" /> is called local if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167020.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167021.png" /> with a local set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167022.png" />-subsystems belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167023.png" />. Theorems which establish the local nature of given abstract classes are called local (cf. [[Mal'tsev local theorems|Mal'tsev local theorems]]).
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An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167024.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167025.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167027.png" />-approximable (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167029.png" />-residual) if, for any predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167030.png" /> (i.e. for any basic predicate as well as for the predicate coinciding with the equality relation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167031.png" />) and for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167033.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167034.png" />, there exists a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167035.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167036.png" /> into some system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167037.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167038.png" /> for which, again, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167039.png" />. Any subsystem of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167040.png" />-approximable system is itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167041.png" />-approximable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167042.png" /> is the class of all finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167043.png" />-systems, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167044.png" />-approximable system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167045.png" /> is called finitely approximable (or residually finite). If an abstract class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167046.png" /> has a unit system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167047.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167048.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167049.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167050.png" />-approximable if and only if it is isomorphically imbeddable in a Cartesian product of systems from the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167051.png" /> [[#References|[3]]]. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167052.png" /> is called residual if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167053.png" />-approximable systems belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167054.png" />. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167055.png" /> is homomorphically closed if it contains, for each of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167056.png" />-systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167057.png" />, also all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167058.png" />-systems that are homomorphic images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167059.png" />. All residual homomorphically-closed classes are local [[#References|[5]]].
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A class of algebraic systems of the same type. All systems of a given type are assumed to be written in a given signature  $  \Omega $
 +
and are called  $  \Omega $-
 +
systems. A class $  \mathfrak K $
 +
of $  \Omega $-
 +
systems is called abstract if, whenever it contains a system $  \mathbf A $,  
 +
it also contains all $  \Omega $-
 +
systems isomorphic to $  \mathbf A $.
  
A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167061.png" />-systems is called (finitely) axiomatizable if there exists a (finite) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167062.png" /> of first-order closed formulas of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167063.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167064.png" /> consists of exactly those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167065.png" />-systems in which all formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167066.png" /> are true. Finitely-axiomatizable classes are also referred to as elementary classes. It has been shown [[#References|[5]]] with the aid of the generalized continuum hypothesis that: 1) a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167067.png" /> of algebraic systems is axiomatizable if and only if it is closed with respect to ultra-products, and its complement (in the class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167068.png" />-systems) is closed with respect to ultra-powers; 2) a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167069.png" /> of algebraic systems is elementary if and only if both it and its complement are closed with respect to ultra-products. The theory of axiomatizable classes of algebraic systems deals with the connection between the structural properties of these classes and the syntactic features of the formal language in which these classes may be specified. Axiomatizable classes which play an especially important role in algebra include varieties (cf. [[Algebraic systems, variety of|Algebraic systems, variety of]]) and quasi-varieties (cf. [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]]), which are local and residual.
+
Let  $  \mathfrak K $
 +
be an abstract class of  $  \Omega $-
 +
systems. One says that an  $  \Omega $-
 +
system  $  \mathbf A $
 +
has a local set of $  \mathfrak K $-
 +
subsystems if there exists an inclusion-directed set $  \{ {\mathbf A _  \alpha  } : {\alpha \in \Lambda } \} $
 +
of subsystems  $  \mathbf A _  \alpha  $
 +
of the system  $  \mathbf A $
 +
which cover  $  \mathbf A $(
 +
i.e. $  \cup _  \alpha  A _  \alpha  = A $)
 +
and which belong to $  \mathfrak K $.  
 +
A class $  \mathfrak K $
 +
is called local if each  $  \Omega $-
 +
system  $  \mathbf A $
 +
with a local set of $  \mathfrak K $-
 +
subsystems belongs to the class $  \mathfrak K $.  
 +
Theorems which establish the local nature of given abstract classes are called local (cf. [[Mal'tsev local theorems|Mal'tsev local theorems]]).
  
In addition to axiomatizability by first-order closed formulas, axiomatizability by special second-order closed formulas is also considered. To the function and predicate signature symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167070.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167071.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167072.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167073.png" />) of a fixed signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167074.png" /> are added predicate variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167075.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167076.png" /> be a quantifier-free formula of the first order, consisting of function and predicate signature symbols, predicate variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167077.png" /> and object variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167078.png" />. A second-order formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167079.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167080.png" /> is some sequence of quantifiers of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167081.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167082.png" /> is known as crypto-universal. The second-order formulas formed from the crypto-universal formulas without free object variables, with the aid of the logical connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167083.png" /> and with subsequent quantification by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167084.png" /> on all free predicate variables encountered in the representations of the crypto-universal formulas, are called Boolean-universal formulas of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167085.png" />. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167087.png" />-systems is called quasi-universal if there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167088.png" /> of Boolean-universal formulas of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167089.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167090.png" /> consists of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167091.png" />-systems in which all formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167092.png" /> are true, and only of such systems. A quasi-universal class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011670/a01167093.png" />-systems is local (Mal'tsev's theorem). A.I. Mal'tsev [[#References|[4]]] gave a more detailed definition of a quasi-universal class.
+
An  $  \Omega $-
 +
system  $  \mathbf A $
 +
is called  $  \mathfrak K $-
 +
approximable (or  $  \mathfrak K $-
 +
residual) if, for any predicate  $  P \in \{ \Omega _ {p} , = \} $(
 +
i.e. for any basic predicate as well as for the predicate coinciding with the equality relation in  $  \mathbf A $)
 +
and for any elements  $  a _ {1} \dots a _ {n} $
 +
in  $  \mathbf A $
 +
for which  $  P ( a _ {1} \dots a _ {n} ) = F $,
 +
there exists a homomorphism  $  \phi :  \mathbf A \rightarrow \mathbf B $
 +
of the system  $  \mathbf A $
 +
into some system  $  \mathbf B $
 +
of the class  $  \mathfrak K $
 +
for which, again,  $  P ( \phi ( a _ {1} ) \dots \phi ( a _ {n} ) ) = F $.
 +
Any subsystem of a  $  \mathfrak K $-
 +
approximable system is itself  $  \mathfrak K $-
 +
approximable. If  $  \mathfrak K $
 +
is the class of all finite  $  \Omega $-
 +
systems, a  $  \mathfrak K $-
 +
approximable system  $  \mathfrak A $
 +
is called finitely approximable (or residually finite). If an abstract class  $  \mathfrak K $
 +
has a unit system  $  E = \langle  \{ e \} , \Omega \rangle $,
 +
an  $  \Omega $-
 +
system  $  \mathbf A $
 +
is  $  \mathfrak K $-
 +
approximable if and only if it is isomorphically imbeddable in a Cartesian product of systems from the class  $  \mathfrak K $[[#References|[3]]]. A class  $  \mathfrak K $
 +
is called residual if all  $  \mathfrak K $-
 +
approximable systems belong to the class  $  \mathfrak K $.
 +
A class  $  \mathfrak K $
 +
is homomorphically closed if it contains, for each of its  $  \Omega $-
 +
systems  $  \mathbf A $,
 +
also all  $  \Omega $-
 +
systems that are homomorphic images of  $  \mathbf A $.
 +
All residual homomorphically-closed classes are local [[#References|[5]]].
 +
 
 +
A class  $  \mathfrak K $
 +
of  $  \Omega $-
 +
systems is called (finitely) axiomatizable if there exists a (finite) set  $  S $
 +
of first-order closed formulas of the signature  $  \Omega $
 +
such that  $  \mathfrak K $
 +
consists of exactly those  $  \Omega $-
 +
systems in which all formulas of  $  S $
 +
are true. Finitely-axiomatizable classes are also referred to as elementary classes. It has been shown [[#References|[5]]] with the aid of the generalized continuum hypothesis that: 1) a class  $  \mathfrak K $
 +
of algebraic systems is axiomatizable if and only if it is closed with respect to ultra-products, and its complement (in the class of all  $  \Omega $-
 +
systems) is closed with respect to ultra-powers; 2) a class  $  \mathfrak K $
 +
of algebraic systems is elementary if and only if both it and its complement are closed with respect to ultra-products. The theory of axiomatizable classes of algebraic systems deals with the connection between the structural properties of these classes and the syntactic features of the formal language in which these classes may be specified. Axiomatizable classes which play an especially important role in algebra include varieties (cf. [[Algebraic systems, variety of|Algebraic systems, variety of]]) and quasi-varieties (cf. [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]]), which are local and residual.
 +
 
 +
In addition to axiomatizability by first-order closed formulas, axiomatizability by special second-order closed formulas is also considered. To the function and predicate signature symbols $  F _ {i} $(
 +
$  i \in I $),  
 +
$  P _ {j} $(
 +
$  j \in I $)  
 +
of a fixed signature $  \Omega $
 +
are added predicate variables $  R _ {1} , R _ {2} ,\dots $.  
 +
Let $  \mathfrak F $
 +
be a quantifier-free formula of the first order, consisting of function and predicate signature symbols, predicate variables $  R _ {1} \dots R _ {s} $
 +
and object variables $  x _ {1} \dots x _ {r} $.  
 +
A second-order formula $  Q  \mathfrak F $,  
 +
where $  Q $
 +
is some sequence of quantifiers of the type $  ( \forall R _ {i} ), ( \exists R _ {i} ) $
 +
or $  ( \forall x _ {k} ) $
 +
is known as crypto-universal. The second-order formulas formed from the crypto-universal formulas without free object variables, with the aid of the logical connectives $  \& , \lor , \rightarrow , \neg $
 +
and with subsequent quantification by $  \forall $
 +
on all free predicate variables encountered in the representations of the crypto-universal formulas, are called Boolean-universal formulas of the signature $  \Omega $.  
 +
A class $  \mathfrak K $
 +
of $  \Omega $-
 +
systems is called quasi-universal if there exists a set $  S $
 +
of Boolean-universal formulas of the signature $  \Omega $
 +
such that $  \mathfrak K $
 +
consists of the $  \Omega $-
 +
systems in which all formulas of $  S $
 +
are true, and only of such systems. A quasi-universal class of $  \Omega $-
 +
systems is local (Mal'tsev's theorem). A.I. Mal'tsev [[#References|[4]]] gave a more detailed definition of a quasi-universal class.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "A general method for obtaining local theorems in group theory"  ''Uchen. Zap. Ivanovsk. Gos. Ped. Inst.'' , '''1''' :  1  (1941)  pp. 3–9  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Model correspondences"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23'''  (1959)  pp. 313–336  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Mal'tsev,  "Some problems in the theory of classes of models" , ''Proc. 4-th All-Union Math. Congress (1961)'' , '''1''' , Leningrad  (1963)  pp. 169–198  (In Russian)  (Transl. in: Amer. Math. Soc. Transl. (2) 83 (1969), 1–48)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.P. Cleave,  "Local properties of systems"  ''J. London Math. Soc.'' , '''44'''  (1969)  pp. 121–130</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "A general method for obtaining local theorems in group theory"  ''Uchen. Zap. Ivanovsk. Gos. Ped. Inst.'' , '''1''' :  1  (1941)  pp. 3–9  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Model correspondences"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23'''  (1959)  pp. 313–336  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Mal'tsev,  "Some problems in the theory of classes of models" , ''Proc. 4-th All-Union Math. Congress (1961)'' , '''1''' , Leningrad  (1963)  pp. 169–198  (In Russian)  (Transl. in: Amer. Math. Soc. Transl. (2) 83 (1969), 1–48)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.P. Cleave,  "Local properties of systems"  ''J. London Math. Soc.'' , '''44'''  (1969)  pp. 121–130</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 16:10, 1 April 2020


A class of algebraic systems of the same type. All systems of a given type are assumed to be written in a given signature $ \Omega $ and are called $ \Omega $- systems. A class $ \mathfrak K $ of $ \Omega $- systems is called abstract if, whenever it contains a system $ \mathbf A $, it also contains all $ \Omega $- systems isomorphic to $ \mathbf A $.

Let $ \mathfrak K $ be an abstract class of $ \Omega $- systems. One says that an $ \Omega $- system $ \mathbf A $ has a local set of $ \mathfrak K $- subsystems if there exists an inclusion-directed set $ \{ {\mathbf A _ \alpha } : {\alpha \in \Lambda } \} $ of subsystems $ \mathbf A _ \alpha $ of the system $ \mathbf A $ which cover $ \mathbf A $( i.e. $ \cup _ \alpha A _ \alpha = A $) and which belong to $ \mathfrak K $. A class $ \mathfrak K $ is called local if each $ \Omega $- system $ \mathbf A $ with a local set of $ \mathfrak K $- subsystems belongs to the class $ \mathfrak K $. Theorems which establish the local nature of given abstract classes are called local (cf. Mal'tsev local theorems).

An $ \Omega $- system $ \mathbf A $ is called $ \mathfrak K $- approximable (or $ \mathfrak K $- residual) if, for any predicate $ P \in \{ \Omega _ {p} , = \} $( i.e. for any basic predicate as well as for the predicate coinciding with the equality relation in $ \mathbf A $) and for any elements $ a _ {1} \dots a _ {n} $ in $ \mathbf A $ for which $ P ( a _ {1} \dots a _ {n} ) = F $, there exists a homomorphism $ \phi : \mathbf A \rightarrow \mathbf B $ of the system $ \mathbf A $ into some system $ \mathbf B $ of the class $ \mathfrak K $ for which, again, $ P ( \phi ( a _ {1} ) \dots \phi ( a _ {n} ) ) = F $. Any subsystem of a $ \mathfrak K $- approximable system is itself $ \mathfrak K $- approximable. If $ \mathfrak K $ is the class of all finite $ \Omega $- systems, a $ \mathfrak K $- approximable system $ \mathfrak A $ is called finitely approximable (or residually finite). If an abstract class $ \mathfrak K $ has a unit system $ E = \langle \{ e \} , \Omega \rangle $, an $ \Omega $- system $ \mathbf A $ is $ \mathfrak K $- approximable if and only if it is isomorphically imbeddable in a Cartesian product of systems from the class $ \mathfrak K $[3]. A class $ \mathfrak K $ is called residual if all $ \mathfrak K $- approximable systems belong to the class $ \mathfrak K $. A class $ \mathfrak K $ is homomorphically closed if it contains, for each of its $ \Omega $- systems $ \mathbf A $, also all $ \Omega $- systems that are homomorphic images of $ \mathbf A $. All residual homomorphically-closed classes are local [5].

A class $ \mathfrak K $ of $ \Omega $- systems is called (finitely) axiomatizable if there exists a (finite) set $ S $ of first-order closed formulas of the signature $ \Omega $ such that $ \mathfrak K $ consists of exactly those $ \Omega $- systems in which all formulas of $ S $ are true. Finitely-axiomatizable classes are also referred to as elementary classes. It has been shown [5] with the aid of the generalized continuum hypothesis that: 1) a class $ \mathfrak K $ of algebraic systems is axiomatizable if and only if it is closed with respect to ultra-products, and its complement (in the class of all $ \Omega $- systems) is closed with respect to ultra-powers; 2) a class $ \mathfrak K $ of algebraic systems is elementary if and only if both it and its complement are closed with respect to ultra-products. The theory of axiomatizable classes of algebraic systems deals with the connection between the structural properties of these classes and the syntactic features of the formal language in which these classes may be specified. Axiomatizable classes which play an especially important role in algebra include varieties (cf. Algebraic systems, variety of) and quasi-varieties (cf. Algebraic systems, quasi-variety of), which are local and residual.

In addition to axiomatizability by first-order closed formulas, axiomatizability by special second-order closed formulas is also considered. To the function and predicate signature symbols $ F _ {i} $( $ i \in I $), $ P _ {j} $( $ j \in I $) of a fixed signature $ \Omega $ are added predicate variables $ R _ {1} , R _ {2} ,\dots $. Let $ \mathfrak F $ be a quantifier-free formula of the first order, consisting of function and predicate signature symbols, predicate variables $ R _ {1} \dots R _ {s} $ and object variables $ x _ {1} \dots x _ {r} $. A second-order formula $ Q \mathfrak F $, where $ Q $ is some sequence of quantifiers of the type $ ( \forall R _ {i} ), ( \exists R _ {i} ) $ or $ ( \forall x _ {k} ) $ is known as crypto-universal. The second-order formulas formed from the crypto-universal formulas without free object variables, with the aid of the logical connectives $ \& , \lor , \rightarrow , \neg $ and with subsequent quantification by $ \forall $ on all free predicate variables encountered in the representations of the crypto-universal formulas, are called Boolean-universal formulas of the signature $ \Omega $. A class $ \mathfrak K $ of $ \Omega $- systems is called quasi-universal if there exists a set $ S $ of Boolean-universal formulas of the signature $ \Omega $ such that $ \mathfrak K $ consists of the $ \Omega $- systems in which all formulas of $ S $ are true, and only of such systems. A quasi-universal class of $ \Omega $- systems is local (Mal'tsev's theorem). A.I. Mal'tsev [4] gave a more detailed definition of a quasi-universal class.

References

[1] A.I. Mal'tsev, "A general method for obtaining local theorems in group theory" Uchen. Zap. Ivanovsk. Gos. Ped. Inst. , 1 : 1 (1941) pp. 3–9 (In Russian)
[2] A.I. Mal'tsev, "Model correspondences" Izv. Akad. Nauk SSSR Ser. Mat. , 23 (1959) pp. 313–336 (In Russian)
[3] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[4] A.I. Mal'tsev, "Some problems in the theory of classes of models" , Proc. 4-th All-Union Math. Congress (1961) , 1 , Leningrad (1963) pp. 169–198 (In Russian) (Transl. in: Amer. Math. Soc. Transl. (2) 83 (1969), 1–48)
[5] P.M. Cohn, "Universal algebra" , Reidel (1981)
[6] J.P. Cleave, "Local properties of systems" J. London Math. Soc. , 44 (1969) pp. 121–130

Comments

The characterization (under the generalized continuum hypothesis) of axiomatizable and elementary classes is due to H.J. Keisler [a1].

Translations of the articles [1], [2] and [4] may (also) be found in [a2], Chapts. 2, 11 and 26, respectively.

The term inductive class is sometimes used instead of "local class" .

References

[a1] H.J. Keisler, "Ultraproducts and elementary classes" Indag. Math. , 23 (1961) pp. 477–495
[a2] A.I. [A.I. Mal'tsev] Mal'cev, , The metamathematics of algebraic systems. Collected papers: 1936 - 1967 , North-Holland (1971)
How to Cite This Entry:
Algebraic systems, class of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_systems,_class_of&oldid=45066
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article