Algebraic hull system

Algebraic envelope systems are a term from the mathematical branch of universal algebra . A shell system is called algebraic if it results from the set of universes of all substructures of an algebraic structure .

Relationship between the envelope system and the envelope operator

For a system of hulls over a basic set the corresponding hull operator is given by: ${\displaystyle {\mathcal {H}}\subseteq {\mathcal {P}}(S)}$ ${\displaystyle S}$ ${\displaystyle C_{\mathcal {H}}}$ ${\displaystyle {\mathcal {P}}(S)}$

${\displaystyle C_{\mathcal {H}}(A)=\bigcap \{H\in {\mathcal {H}}|H\supseteq A\}}$( ). ${\displaystyle A\subseteq S}$

The hull operator assigns the smallest superset from the hull system to a subset of S.

Characterization via finiteness condition

The algebraicity of a hull system can be characterized as follows without recourse to algebraic structures: The hull system and the hull operator are called algebraic if the following finiteness condition is met: ${\displaystyle {\mathcal {H}}}$ ${\displaystyle C_{\mathcal {H}}}$

If and , then there already exists a finite subset such that . ${\displaystyle A\subseteq S}$ ${\displaystyle x\in C_{\mathcal {H}}(A)}$ ${\displaystyle F\subseteq A}$ ${\displaystyle x\in C_{\mathcal {H}}(F)}$

That means:

It is always

${\displaystyle C_{\mathcal {H}}(A)=\bigcup \{C_{\mathcal {H}}(F):F\subseteq A\land |F|<\infty \}}$( ). ${\displaystyle A\subseteq S}$

In logic this property is called compactness .

This property applies to every shell system that is given by the substructures of an algebraic structure, because an element of the structure lies in the product of a subset of the structure if there is a term consisting of the (finite-place) connections of the structure and elements of the Is a subset whose value is, and a term can only use a finite number of such elements. Conversely, a corresponding algebraic structure can be defined for an envelope system with the above property by setting a link for each and and as defined above by and for other tuples (which does not occur for), for example . ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle A\subseteq S}$ ${\displaystyle x}$ ${\displaystyle F=\left\{f_{1},\ldots ,f_{n}\right\}}$ ${\displaystyle v\colon S^{n}\mapsto S}$ ${\displaystyle v(f_{1},\ldots ,f_{n})=x}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle n=0}$ ${\displaystyle v(x_{1},\ldots ,x_{n})=x_{1}}$

Characterization via inductance

A set of sets is called inductive if for every non-empty subset that is linearly ordered in ascending order with respect to the inclusion relation, the union set belongs to . This is equivalent to the union of every non-empty subset of turn belonging to, which is directed with respect to the inclusion relation . The reverse direction follows a fortiori , the execution results from transfinite induction over all cardinal numbers: Consider a finite directed set as the induction start, this has a maximum, which makes the statement trivial. So now be a directed subset with infinite cardinality . can be represented as the union of an ascending chain of subsets of smaller cardinality. For this choose a numbering , then there is a union of the images for each ordinal number . Since for infinite sets the set of all finite subsets has the same cardinality as the set itself and thus each can be added to a directed subset without exceeding the cardinality , it can even be used as a union of an ascending chain of directed subsets of smaller cardinality. For this the assertion is shown by induction hypothesis and it results for all cardinal numbers. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {K}}\subseteq {\mathcal {T}}}$ ${\displaystyle \textstyle \bigcup {\mathcal {K}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {K}}\subseteq {\mathcal {T}}}$ ${\displaystyle \kappa }$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle f\colon \kappa \to {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle f(\alpha )}$ ${\displaystyle \alpha <\kappa }$ ${\displaystyle f(\alpha )}$ ${\displaystyle \kappa }$ ${\displaystyle {\mathcal {K}}}$

Schmidt's theorem

A sentence by Jürgen Schmidt (1918–1980) results from the above , which states that the inductance for a shell system is equivalent to the algebraicity.

Because algebraicity obviously implies inductance directly. Conversely, consider the directed set for a hull system and a (it is directed, there ). It consists of elements of the hull system, so its union is also an element of the hull system, thus and the algebraicity is shown. Note that the proof of the latter implication is based on the axiom of choice due to the above use of certain theorems about infinite sets . ${\displaystyle {\mathcal {H}}\subseteq {\mathcal {P}}(S)}$ ${\displaystyle A\subseteq S}$ ${\displaystyle \left\{C_{\mathcal {H}}(F)\mid F\subseteq A\land |F|<\infty \right\}}$ ${\displaystyle C_{\mathcal {H}}(F)\cup C_{\mathcal {H}}(G)=C_{\mathcal {H}}(F\cup G)}$ ${\displaystyle U}$ ${\displaystyle U=C_{\mathcal {H}}(A)}$

Examples

Two simple examples can be used to check the relationship between algebraicity and inductance formulated by the sentence.

One possible barrier system is the whole power set, . In this case the hull operator is identity. Since every subset of is the union of its finite subsets, the hull operator and the hull system are algebraic. In fact, in this case the envelope system is also inductive. ${\displaystyle {\mathcal {H}}={\mathcal {P}}(S)}$ ${\displaystyle S}$

Another barrier system consists of the assumed to be infinite amount and all finite subsets . In this case, finite subsets are mapped to themselves by the envelope operator, whereas infinite subsets are mapped to whole . For an infinite real subset of the finiteness condition is therefore not fulfilled, and the hull system is therefore not algebraic. In fact, it is not inductive either, an ascending chain of finite sets that does not fully exhaust is a counterexample for this. ${\displaystyle S}$ ${\displaystyle {\mathcal {H}}=\{S\}\cup \{A:A\subseteq S\land |A|<\infty \}}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

literature

Original work

• Jürgen Schmidt: About the role of transfinite conclusions in a general ideal theory . In: Math . tape 7 , 1952, pp. 165-182 ( MR0047628 ).
• Jürgen Schmidt: Some basic terms and sentences from the theory of the envelope operators. In: Report on the mathematicians' conference in Berlin . Deutscher Verlag der Wissenschaften, Berlin January 1953, p. 21-48 ( MR0069802 ).

Monographs

• Paul Moritz Cohn : Universal Algebra (=  Mathematics and Its Applications . Volume 6 ). Revised edition. D. Reidel Publishing, Dordrecht, Boston 1981, ISBN 90-277-1213-1 .
• Th. Ihringer : General Algebra (=  Teubner Study Book ). Teubner Verlag, Stuttgart 1988, ISBN 3-519-02083-1 .
• Heinrich Werner: Introduction to general algebra (=  BI university pocket book . Volume 120 ). Bibliographical Institute, Mannheim / Vienna / Zurich 1978, ISBN 3-411-00120-8 .

References and footnotes

1. ^ Bjarni Jónsson : Topics in Universal Algebra . Springer, Berlin 1972, ISBN 3-540-05722-6 , pp. 91 .
2. ↑ Ihringer: p. 37.
3. ↑ Ihringer does not speak of algebraic envelope systems in his presentation , but focuses them solely on the concept of inductance.
4. ↑ Stanley Burris, HP Sankappanavar: A Course in Universal Algebra . 1981, p. 24 ( math.uwaterloo.ca [PDF; 1.6 MB ]).
5. ^ Schmidt: Math. Nachr . tape 7 , 1952, pp. 174 .
6. ^ Günter Bruns: A lemma on directed sets and chains . In: Archives of Mathematics . tape 18 , no. 6 . Birkhäuser, 1967, ISSN  0003-889X , p. 561-563 , doi : 10.1007 / BF01898858 .
7. ^ Schmidt: Math. Nachr . tape 7 , 1952, pp. 172 .
8. ^ Schmidt: Report on the mathematicians' conference in Berlin . January 1953, p. 25 .
9. ^ Cohn: pp. 45, 397.
10. ↑ Schmidt describes the theorem in the articles from 1952 and 1953 as the main clause on algebraic envelope systems. This term is not used in today's literature on universal algebra. Heinrich Werner gives an introduction to general algebra. P. 32, a sentence that essentially corresponds to Schmidt's sentence and is nevertheless not assigned to Jürgen Schmidt, but is named as a result by Birkhoff-Frink from 1948.