Universal algebra

The universal algebra (including general algebra ) is a branch of mathematics , more precisely the algebra that deals with general algebraic structures and their homomorphisms concerned and certain generalizations.

While in abstract algebra and its respective sub-areas such as group theory , ring theory and body theory algebraic structures with certain fixed links with fixed properties are examined, universal algebra deals with structures in general, i.e. with structures with any links and any definable properties. Group theory, for example, talks about groups in general , whereas for universal algebra groups are just one example of one type of algebraic structure. Universal algebra is related to model theory , a branch of mathematical logic that deals with the relationship between structures and the logical formulas that describe them. The model theory of equation logic is of central interest . The lattice theory has applications in universal algebra. The category theory provides a more general approach of the universal algebra can be viewed from the. The description of structures is reduced to the behavior of their structure-preserving mappings under concatenation , in the case of the universal algebra of homomorphisms.

Basic concepts

The fundamental concept of universal algebra is that of algebraic structure. An algebraic structure is a quantity called the amount of carrier , provided with a family of links may be of different arities , wherein each an arbitrary natural number is. Constants can be represented formally by 0-digit links. A group, for example, is an algebraic structure with a two-digit link, the respective group multiplication. A ring, on the other hand, has two two-digit links, the respective addition and the respective multiplication. ${\ displaystyle A}$ ${\ displaystyle f_ {i} \ colon A ^ {n_ {i}} \ to A}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle n_ {i}}$

When defining a group or a ring and many other structures, it is also required that the links meet certain properties, such as the associative law . Classes of algebraic structures that fulfill certain properties that are given by logical formulas are therefore a natural object of investigation . In many cases, the simple equation logic is sufficient. The group axioms, for example, can be formulated in this - with the addition of one-digit or zero-digit links for the formation of the inverse and the neutral element. This logic has the pleasant property that every substructure of an algebraic structure, i.e. H. a subset, as long as the links are still well-defined, satisfies the same equation-logic formulas. Those classes form a special case of the elementary classes of structures examined in classical model theory , which are axiomatized by formulas of first-order predicate logic .

A homomorphism between two algebraic structures and with links or with the same arity is a mapping with the property that for each and for all the equation ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle f_ {i}}$ ${\ displaystyle g_ {i}}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle p \ colon A \ to B}$ ${\ displaystyle i}$ ${\ displaystyle a_ {1}, \ ldots, a_ {n_ {i}}}$

{\ displaystyle p (f_ {i} (a_ {1}, \ ldots, a_ {n_ {i}})) = g_ {i} (p (a_ {1}), \ ldots, p (a_ {n_ { i}}))}

applies. Every bijective homomorphism on an algebraic structure is an isomorphism . With the homomorphisms as morphisms , the algebraic structures form a category , so that the usual general category-theoretical terms can be used.

Generalizations

In addition to simple algebraic structures, various generalizations are also considered, to which certain sentences can sometimes be transferred, for example:

Partial algebraic structures , the links, do not have to be defined for all combinations of parameters.

Heterogeneous algebras and partially heterogeneous algebras, instead of one carrier set, there are several carrier sets on which links are defined.

Relational structures , instead of functions, any relations are allowed, these are typical objects of investigation in model theory.

Infinitary structures , the connections can have infinite arity .

Topological algebraic structures , the structures are additionally provided with a topological structure , with respect to which the operations are continuous.

history

The British mathematician Alfred North Whitehead published his Treatise on Universal Algebra in 1898 . In this work he spoke in a general way of operations and equations, by universal algebra, but by universal algebra he meant only the study of structures with two internal connections ( i.e. two magma structures , called addition and multiplication), with different possible ones additional properties, and possibly a kind of generalized graduation . In contrast, he did not achieve general results in universal algebra. Garrett Birkhoff delivered these for the first time in 1935 . From 1941 onwards, Anatoli Ivanovich Malzew applied the early theoretical model results, which he had brought into general, modern form, to universal algebra for the first time.

literature

Garrett Birkhoff : Lattice Theory . 3. Edition. American Mathematical Society, Providence, Rhode Island 1979.
Stanley Burris, HP Sankappanavar: A Course in Universal Algebra . Ed .: Natural Sciences and Engineering Research Council Canada (= Graduate texts in mathematics . No. 78 ). Ottawa, Ontario, Canada 2000 ( math.uwaterloo.ca [PDF; 15.5 MB ]).
George Grätzer: Universal Algebra . Van Nostrand, Princeton (NJ) 1968, ISBN 978-0-387-77486-2 , doi : 10.1007 / 978-0-387-77487-9 .
Thomas Ihringer: General Algebra . With an appendix on Universal Coalgebra by HP Gumm (= Berlin study series on mathematics . Volume 10 ). Heldermann, Lemgo 2003, ISBN 3-88538-110-9 .
Anatolij Ivanovič Mal'cev: The Metamathematics of Algebraic Systems . Collected Papers: 1936–1967 (= Studies in logic and the foundations of mathematics . Volume 66 ). North-Holland, Amsterdam 1971 (translated from Russian by Benjamin Franklin Wells).
Heinrich Werner: Introduction to general algebra (= BI university pocket books . Volume 120 ). Bibliographisches Institut, Mannheim u. a. 1978, ISBN 3-411-00120-8 .

Web links

Lev Aleksandrovich Skornyakov: Universal algebra . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Alex Sakharov, Matt Insall: Universal Algebra . In: MathWorld (English).
universal algebra , entry in the nLab . (English)

Individual evidence

^ Heinrich Werner: Review of the book Equational logic by Walter Taylor . In: The Journal of Symbolic Logic . tape 47 , no. 2 , 1982, p. 450 , doi : 10.2307 / 2273161 , JSTOR : 2273161 .
^ Alfred North Whitehead : A Treatise on Universal Algebra . with Applications. Cambridge University Press, Cambridge 1898 ( projecteuclid.org ).
↑ ^a ^b George Grätzer: Universal Algebra. P. Vii.
^ Lev Aleksandrovich Skornyakov: Universal algebra.
↑ The general variants of the Löwenheim-Skolem theorem, the compactness theorem and the completeness theorem , which allow for uncountable signatures, go back to him, see Juliette Kennedy: Kurt Gödel. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . .
↑ George Grätzer: Universal Algebra. S. viii.

[1] Heinrich Werner: Review of the book Equational logic by Walter Taylor . In: The Journal of Symbolic Logic . tape 47 , no. 2 , 1982, p. 450 , doi : 10.2307 / 2273161 , JSTOR : 2273161 .

[2] Alfred North Whitehead : A Treatise on Universal Algebra . with Applications. Cambridge University Press, Cambridge 1898 ( projecteuclid.org ).

[GrätzerVII-3] George Grätzer: Universal Algebra. P. Vii.

[4] Lev Aleksandrovich Skornyakov: Universal algebra.

[5] The general variants of the Löwenheim-Skolem theorem, the compactness theorem and the completeness theorem , which allow for uncountable signatures, go back to him, see Juliette Kennedy: Kurt Gödel. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . .

[6] George Grätzer: Universal Algebra. S. viii.