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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 [[#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]]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Horn,   "On sentences which are true of direct unions of algebras"  ''J. Symbolic Logic'' , '''16'''  (1951)  pp. 14–21</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Isbell,   "General functional semantics, I"  ''Amer. J. Math.'' , '''94'''  (1972)  pp. 535–596</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.I. [A.I. Mal'tsev] Mal'cev,  , ''The metamathematics of algebraic systems. Collected papers: 1936 - 1967'' , North-Holland  (1971)</TD></TR></table>