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A class of algebraic systems (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116801.png" />-systems) axiomatized by special formulas of a first-order logical language, called quasi-identities or conditional identities, of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116802.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116803.png" /></td> </tr></table>
+
A class of algebraic systems ( $  \Omega $-
 +
systems) axiomatized by special formulas of a first-order logical language, called quasi-identities or conditional identities, of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116804.png" /></td> </tr></table>
+
$$
 +
( \forall x _ {1} ) \dots ( \forall x _ {s} )
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116805.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116806.png" /> are terms of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116807.png" /> in the object variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116808.png" />. By virtue of Mal'tsev's theorem [[#References|[1]]] a quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a0116809.png" /> of algebraic systems of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168010.png" /> can also be defined as an abstract class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168011.png" />-systems containing the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168012.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168013.png" />, and which is closed with respect to subsystems and filtered products [[#References|[1]]], [[#References|[2]]]. An axiomatizable class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168014.png" />-systems is a quasi-variety if and only if it contains the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168015.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168016.png" /> and is closed with respect to subsystems and Cartesian products. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168017.png" /> is a quasi-variety of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168018.png" />, the subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168019.png" /> of systems of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168020.png" /> that are isomorphically imbeddable into suitable systems of some quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168021.png" /> with signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168022.png" />, is itself a quasi-variety. Thus, the class of semi-groups imbeddable into groups is a quasi-variety; the class of associative rings without zero divisors imbeddable into associative skew-fields is also a quasi-variety.
+
$$
 +
[ P _ {1} ( f _ {1} ^ { (1) } \dots f _ {m _ {1}  } ^ { (1) } )  \& {} \dots
 +
\&  P _ {k} ( f _ {1} ^ { (k) } \dots f _ {m _ {k}  } ^ { (k) } )
 +
{} \rightarrow
 +
$$
  
A quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168023.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168024.png" /> is called finitely definable (or, is said to have a finite basis of quasi-identities) if there exists a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168025.png" /> of quasi-identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168027.png" /> consists of only those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168028.png" />-systems in which all the formulas from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168029.png" /> are true. For instance, the quasi-variety of all semi-groups with cancellation is defined by the two quasi-identities
+
$$
 +
\rightarrow \
 +
{} P _ {0} ( f _ {1} ^ { (0) } \dots f _ {m _ {0}  } ^ { (0) } ) ] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168030.png" /></td> </tr></table>
+
where  $  P _ {0} \dots P _ {k} \in \Omega _ {p} \cup \{ = \} $,
 +
and  $  f _ {1} ^ { (0) } \dots f _ {m _ {k}  } ^ { (k) } $
 +
are terms of the signature  $  \Omega $
 +
in the object variables  $  x _ {1} \dots x _ {s} $.
 +
By virtue of Mal'tsev's theorem [[#References|[1]]] a quasi-variety  $  \mathfrak K $
 +
of algebraic systems of signature  $  \Omega $
 +
can also be defined as an abstract class of  $  \Omega $-
 +
systems containing the unit  $  \Omega $-
 +
system  $  E $,
 +
and which is closed with respect to subsystems and filtered products [[#References|[1]]], [[#References|[2]]]. An axiomatizable class of  $  \Omega $-
 +
systems is a quasi-variety if and only if it contains the unit  $  \Omega $-
 +
system  $  E $
 +
and is closed with respect to subsystems and Cartesian products. If  $  \mathfrak K $
 +
is a quasi-variety of signature  $  \Omega $,
 +
the subclass  $  \mathfrak K _ {1} $
 +
of systems of  $  \mathfrak K $
 +
that are isomorphically imbeddable into suitable systems of some quasi-variety  $  \mathfrak K  ^  \prime  $
 +
with signature  $  \Omega  ^  \prime  \supseteq \Omega $,
 +
is itself a quasi-variety. Thus, the class of semi-groups imbeddable into groups is a quasi-variety; the class of associative rings without zero divisors imbeddable into associative skew-fields is also a quasi-variety.
 +
 
 +
A quasi-variety  $  \mathfrak K $
 +
of signature  $  \Omega $
 +
is called finitely definable (or, is said to have a finite basis of quasi-identities) if there exists a finite set  $  S $
 +
of quasi-identities of  $  \Omega $
 +
such that  $  \mathfrak K $
 +
consists of only those  $  \Omega $-
 +
systems in which all the formulas from the set  $  S $
 +
are true. For instance, the quasi-variety of all semi-groups with cancellation is defined by the two quasi-identities
 +
 
 +
$$
 +
zx =zy  \rightarrow  x = y ,\  xz = yz  \rightarrow  x = y ,
 +
$$
  
 
and is therefore finitely definable. On the other hand, the quasi-variety of semi-groups imbeddable into groups has no finite basis of quasi-identities [[#References|[1]]], [[#References|[2]]].
 
and is therefore finitely definable. On the other hand, the quasi-variety of semi-groups imbeddable into groups has no finite basis of quasi-identities [[#References|[1]]], [[#References|[2]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168031.png" /> be an arbitrary (not necessarily abstract) class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168032.png" />-systems; the smallest quasi-variety containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168033.png" /> is said to be the implicative closure of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168034.png" />. It consists of subsystems of isomorphic copies of filtered products of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168035.png" />-systems of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168037.png" /> is the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168038.png" />-system. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168039.png" /> is the implicative closure of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168041.png" />-systems, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168042.png" /> is called a generating class of the quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168043.png" />. A quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168044.png" /> is generated by one system if and only if for any two systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168047.png" /> there exists in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168048.png" /> a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168049.png" /> containing subsystems isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168051.png" /> [[#References|[1]]]. Any quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168052.png" /> containing systems other than one-element systems has free systems of any rank, which are at the same time free systems in the equational closure of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168053.png" />. The quasi-varieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168054.png" />-systems contained in some given quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168055.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168056.png" /> constitute a complete lattice with respect to set-theoretic inclusion. The atoms of the lattice of all quasi-varieties of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168057.png" /> are called minimal quasi-varieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168058.png" />. A minimal quasi-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168059.png" /> is generated by any one of its non-unit systems. Every quasi-variety with a non-unit system contains at least one minimal quasi-variety. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168060.png" /> is a quasi-variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168061.png" />-systems of finite signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168062.png" />, all its sub-quasi-varieties constitute a groupoid with respect to the Mal'tsev <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011680/a01168063.png" />-multiplication [[#References|[3]]].
+
Let $  \mathfrak K $
 +
be an arbitrary (not necessarily abstract) class of $  \Omega $-
 +
systems; the smallest quasi-variety containing $  \mathfrak K $
 +
is said to be the implicative closure of the class $  \mathfrak K $.  
 +
It consists of subsystems of isomorphic copies of filtered products of $  \Omega $-
 +
systems of the class $  \mathfrak K \cup \{ E \} $,  
 +
where $  E $
 +
is the unit $  \Omega $-
 +
system. If $  \mathfrak K $
 +
is the implicative closure of a class $  \mathfrak A $
 +
of $  \Omega $-
 +
systems, $  \mathfrak A $
 +
is called a generating class of the quasi-variety $  \mathfrak K $.  
 +
A quasi-variety $  \mathfrak K $
 +
is generated by one system if and only if for any two systems $  \mathbf A $,  
 +
$  \mathbf B $
 +
of $  \mathfrak K $
 +
there exists in the class $  \mathfrak K $
 +
a system $  \mathbf C $
 +
containing subsystems isomorphic to $  \mathbf A $
 +
and $  \mathbf B $[[#References|[1]]]. Any quasi-variety $  \mathfrak K $
 +
containing systems other than one-element systems has free systems of any rank, which are at the same time free systems in the equational closure of the class $  \mathfrak K $.  
 +
The quasi-varieties of $  \Omega $-
 +
systems contained in some given quasi-variety $  \mathfrak K $
 +
of signature $  \Omega $
 +
constitute a complete lattice with respect to set-theoretic inclusion. The atoms of the lattice of all quasi-varieties of signature $  \Omega $
 +
are called minimal quasi-varieties of $  \Omega $.  
 +
A minimal quasi-variety $  \mathfrak M $
 +
is generated by any one of its non-unit systems. Every quasi-variety with a non-unit system contains at least one minimal quasi-variety. If $  \mathfrak K $
 +
is a quasi-variety of $  \Omega $-
 +
systems of finite signature $  \Omega $,  
 +
all its sub-quasi-varieties constitute a groupoid with respect to the Mal'tsev $  \mathfrak K $-
 +
multiplication [[#References|[3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Multiplication of classes of algebraic systems"  ''Siberian Math. J.'' , '''8''' :  2  (1967)  pp. 254–267  ''Sibirsk Mat. Zh.'' , '''8''' :  2  (1967)  pp. 346–365</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,  "Multiplication of classes of algebraic systems"  ''Siberian Math. J.'' , '''8''' :  2  (1967)  pp. 254–267  ''Sibirsk Mat. Zh.'' , '''8''' :  2  (1967)  pp. 346–365</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 16:10, 1 April 2020


A class of algebraic systems ( $ \Omega $- systems) axiomatized by special formulas of a first-order logical language, called quasi-identities or conditional identities, of the form

$$ ( \forall x _ {1} ) \dots ( \forall x _ {s} ) $$

$$ [ P _ {1} ( f _ {1} ^ { (1) } \dots f _ {m _ {1} } ^ { (1) } ) \& {} \dots \& P _ {k} ( f _ {1} ^ { (k) } \dots f _ {m _ {k} } ^ { (k) } ) {} \rightarrow $$

$$ \rightarrow \ {} P _ {0} ( f _ {1} ^ { (0) } \dots f _ {m _ {0} } ^ { (0) } ) ] , $$

where $ P _ {0} \dots P _ {k} \in \Omega _ {p} \cup \{ = \} $, and $ f _ {1} ^ { (0) } \dots f _ {m _ {k} } ^ { (k) } $ are terms of the signature $ \Omega $ in the object variables $ x _ {1} \dots x _ {s} $. By virtue of Mal'tsev's theorem [1] a quasi-variety $ \mathfrak K $ of algebraic systems of signature $ \Omega $ can also be defined as an abstract class of $ \Omega $- systems containing the unit $ \Omega $- system $ E $, and which is closed with respect to subsystems and filtered products [1], [2]. An axiomatizable class of $ \Omega $- systems is a quasi-variety if and only if it contains the unit $ \Omega $- system $ E $ and is closed with respect to subsystems and Cartesian products. If $ \mathfrak K $ is a quasi-variety of signature $ \Omega $, the subclass $ \mathfrak K _ {1} $ of systems of $ \mathfrak K $ that are isomorphically imbeddable into suitable systems of some quasi-variety $ \mathfrak K ^ \prime $ with signature $ \Omega ^ \prime \supseteq \Omega $, is itself a quasi-variety. Thus, the class of semi-groups imbeddable into groups is a quasi-variety; the class of associative rings without zero divisors imbeddable into associative skew-fields is also a quasi-variety.

A quasi-variety $ \mathfrak K $ of signature $ \Omega $ is called finitely definable (or, is said to have a finite basis of quasi-identities) if there exists a finite set $ S $ of quasi-identities of $ \Omega $ such that $ \mathfrak K $ consists of only those $ \Omega $- systems in which all the formulas from the set $ S $ are true. For instance, the quasi-variety of all semi-groups with cancellation is defined by the two quasi-identities

$$ zx =zy \rightarrow x = y ,\ xz = yz \rightarrow x = y , $$

and is therefore finitely definable. On the other hand, the quasi-variety of semi-groups imbeddable into groups has no finite basis of quasi-identities [1], [2].

Let $ \mathfrak K $ be an arbitrary (not necessarily abstract) class of $ \Omega $- systems; the smallest quasi-variety containing $ \mathfrak K $ is said to be the implicative closure of the class $ \mathfrak K $. It consists of subsystems of isomorphic copies of filtered products of $ \Omega $- systems of the class $ \mathfrak K \cup \{ E \} $, where $ E $ is the unit $ \Omega $- system. If $ \mathfrak K $ is the implicative closure of a class $ \mathfrak A $ of $ \Omega $- systems, $ \mathfrak A $ is called a generating class of the quasi-variety $ \mathfrak K $. A quasi-variety $ \mathfrak K $ is generated by one system if and only if for any two systems $ \mathbf A $, $ \mathbf B $ of $ \mathfrak K $ there exists in the class $ \mathfrak K $ a system $ \mathbf C $ containing subsystems isomorphic to $ \mathbf A $ and $ \mathbf B $[1]. Any quasi-variety $ \mathfrak K $ containing systems other than one-element systems has free systems of any rank, which are at the same time free systems in the equational closure of the class $ \mathfrak K $. The quasi-varieties of $ \Omega $- systems contained in some given quasi-variety $ \mathfrak K $ of signature $ \Omega $ constitute a complete lattice with respect to set-theoretic inclusion. The atoms of the lattice of all quasi-varieties of signature $ \Omega $ are called minimal quasi-varieties of $ \Omega $. A minimal quasi-variety $ \mathfrak M $ is generated by any one of its non-unit systems. Every quasi-variety with a non-unit system contains at least one minimal quasi-variety. If $ \mathfrak K $ is a quasi-variety of $ \Omega $- systems of finite signature $ \Omega $, all its sub-quasi-varieties constitute a groupoid with respect to the Mal'tsev $ \mathfrak K $- multiplication [3].

References

[1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[2] P.M. Cohn, "Universal algebra" , Reidel (1981)
[3] A.I. Mal'tsev, "Multiplication of classes of algebraic systems" Siberian Math. J. , 8 : 2 (1967) pp. 254–267 Sibirsk Mat. Zh. , 8 : 2 (1967) pp. 346–365

Comments

In the Western literature, quasi-identities are commonly called Horn sentences (cf. [a1]). For a categorical treatment of quasi-varieties, see [a3]; for their finitary analogue, see [a2]. Mal'tsev's article [a3] may also be found in [a4] as Chapt. 32.

References

[a1] A. Horn, "On sentences which are true of direct unions of algebras" J. Symbolic Logic , 16 (1951) pp. 14–21
[a2] J.R. Isbell, "General functional semantics, I" Amer. J. Math. , 94 (1972) pp. 535–596
[a3] O. Keane, "Abstract Horn theories" F.W. Lawvere (ed.) C. Maurer (ed.) C. Wraith (ed.) , Model theory and topoi , Lect. notes in math. , 445 , Springer (1975) pp. 15–50
[a4] 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, quasi-variety of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_systems,_quasi-variety_of&oldid=45067
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article