Difference between revisions of "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 [[#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 | + | 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) |
Algebraic systems, quasi-variety of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_systems,_quasi-variety_of&oldid=19285