Algebraic systems, quasi-variety of

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

where , and are terms of the signature in the object variables . By virtue of Mal'tsev's theorem [1] a quasi-variety of algebraic systems of signature can also be defined as an abstract class of -systems containing the unit -system , and which is closed with respect to subsystems and filtered products [1], [2]. An axiomatizable class of -systems is a quasi-variety if and only if it contains the unit -system and is closed with respect to subsystems and Cartesian products. If is a quasi-variety of signature , the subclass of systems of that are isomorphically imbeddable into suitable systems of some quasi-variety with signature , 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 of signature is called finitely definable (or, is said to have a finite basis of quasi-identities) if there exists a finite set of quasi-identities of such that consists of only those -systems in which all the formulas from the set are true. For instance, the quasi-variety of all semi-groups with cancellation is defined by the two quasi-identities

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 be an arbitrary (not necessarily abstract) class of -systems; the smallest quasi-variety containing is said to be the implicative closure of the class . It consists of subsystems of isomorphic copies of filtered products of -systems of the class , where is the unit -system. If is the implicative closure of a class of -systems, is called a generating class of the quasi-variety . A quasi-variety is generated by one system if and only if for any two systems , of there exists in the class a system containing subsystems isomorphic to and [1]. Any quasi-variety 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 . The quasi-varieties of -systems contained in some given quasi-variety of signature constitute a complete lattice with respect to set-theoretic inclusion. The atoms of the lattice of all quasi-varieties of signature are called minimal quasi-varieties of . A minimal quasi-variety 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 is a quasi-variety of -systems of finite signature , all its sub-quasi-varieties constitute a groupoid with respect to the Mal'tsev -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)