Algebraic systems, quasi-variety of

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].

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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.

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