# Algebraic systems, quasi-variety of

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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 [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