Universal algebra

Universal algebra is a branch of mathematics that studies the general properties of algebraic systems , looking for common features between algebraic constructions such as groups, rings, modules, lattices, introducing the concepts inherent to all of them and the statements and results common to all of them. It is a section that occupies an intermediate position between mathematical logic and general algebra , as it implements the apparatus of mathematical logic as applied to general algebraic structures.

Content

1 History
2 Algebraic systems, algebras and models
3 Basic designs
4 Varieties
5 Free Algebras
6 Special Algebras
7 Categories of Algebraic Systems
8 Applications
9 notes
10 Literature

History

The first mention of a branch of mathematics with this name refers to Alfred Whitehead (his “Treatise on Universal Algebra, with Applications” ^{[1] was} published in 1898 ) ^[2] , but the emergence of a distinguished discipline that studies algebraic structures as arbitrary sets with arbitrary sets of operations and relationships associated with the work of Garrett Birkhoff in 1935 ^[3] ^[4] , in the framework of the work on the theory of lattices drew attention to a series of parallel structures used in the theory of groups and rings : homomorphisms , quotient and faktorkol tsa , normal subgroups and two-sided ideals . Birkhoff’s works did not evoke published responses and development for some time, but the 1940s saw the emergence of a certain “folklore” associated with such a universal approach to algebra, in particular, the approach was described in the lectures of the late 1940s, delivered by Philip Hall Hall ) at the University of Cambridge ^[2] .

The next step to creating universal algebra as a branch of mathematics is the work of Alfred Tarski on model theory and Kenjiro Soda on algebras with binary operations , as well as the works of Leon Genkin ^[5] , Anatoly Maltsev ^[6] , Abraham Robinson ^[7] , Bjarni Jounsson ( Isl . Bjarni Jónsson ) ^[8] , who drew attention to the effectiveness of applying the apparatus of mathematical logic used in the framework of model theory , which was under construction in those years, to the study of algebraic systems as structures that generalize models and algebras. At the same time, the work of Maltsev 1941 ^{[9] was} marked as anticipating the logical approach to universal algebra, but did not receive feedback and timely development due to the war , and Tarski's lecture at the International Congress of Mathematicians in 1950 was a starting point for the second period of the development of the section ^[10] .

Since the late 1950s, a direction has been developed that studies free algebras , primarily thanks to the work of Edward Marczewski and the subsequent series of more than fifty articles by Polish mathematicians in this direction ^[11] . In the mid-1950s, Philip Higgins introduced and studied multioperator groups ^[12] ^[13] as structures in which the notion of a commutant can be generalized and any congruence can be decomposed into adjacent classes by ideals (by analogy with the corresponding properties of a normal subgroup and two-sided ideal rings), later special classes of multioperator groups (multioperator rings and algebras) were also studied.

Since the beginning of the 1960s, the theory of quasivarieties has been developing and questions of their connection with axiomatizable classes of algebraic systems (Maltsev, Gorbunov ), the most rapidly developing direction of the beginning - the mid-1970s, began to study congruence varieties (Bjarni Jounsson, Gretzer).

By 1968, the bibliography on universal algebra totaled more than 1 thousand articles, by 1980 - more than 5 thousand; in the period from 1976 to 1988, 2 thousand works were published ^[14] .

In the second half of the 1970s, applications of universal algebra in computer science appeared - the theory of abstract data types , the theory of database management systems ^[15] , applications are mainly built around the concept of multi-sorted algebras . Among the main areas that were most actively developed in the 1980s and 1990s ^[16] are the theory of quasivarieties, the theory of commutators for congruence varieties, and the theory of natural duality. In the 2000s, a separate direction was intensively developed: universal algebraic geometry , generalizing classical algebraic geometry working with algebraic fields to wider classes of algebraic systems ^[17] .

Algebraic systems, algebras and models

The basic object of the study of the section, the algebraic system, is an arbitrary nonempty set with a given (possibly infinite) set of finite operations on it and finite relations: ${\mathfrak {A}}=\langle A,\;F,\;R\rangle$ ${\ displaystyle {\ mathfrak {A}} = \ langle A, \; F, \; R \ rangle}$ , $F=\langle f_{1}\colon A^{n_{1}}\to A,\;\ldots f_{i}\colon A^{n_{i}}\to A,\;\ldots \rangle$ ${\ displaystyle F = \ langle f_ {1} \ colon A ^ {n_ {1}} \ to A, \; \ ldots f_ {i} \ colon A ^ {n_ {i}} \ to A, \; \ ldots \ rangle}$ , $R=\langle r_{1}\subseteq A^{m_{1}},\;\ldots r_{i}\subseteq A^{m_{i}},\;\ldots \rangle$ ${\ displaystyle R = \ langle r_ {1} \ subseteq A ^ {m_ {1}}, \; \ ldots r_ {i} \ subseteq A ^ {m_ {i}}, \; \ ldots \ rangle}$ . A bunch of $A$ ${\ displaystyle A}$ in this case is called the carrier (or the main set ) of the system, a set of functional and predicate symbols with their arities $\langle F,\;R,\;\langle n_{1},\;\ldots n_{i},\;\ldots \rangle ,\;\langle m_{1}\ldots m_{i},\;\ldots \rangle \rangle$ ${\ displaystyle \ langle F, \; R, \; \ langle n_ {1}, \; \ ldots n_ {i}, \; \ ldots \ rangle, \; \ langle m_ {1} \ ldots m_ {i} , \; \ ldots \ rangle \ rangle}$ - her signature . A system with an empty set of relations is called universal algebra (in the context of the subject, more often simply an algebra ), and with an empty set of operations it is called a model ^[18] or a system of relations , a relational system ^[19] .

All basic general algebraic structures fit into this abstraction, for example, a partially ordered set — a relational system endowed with a binary relation of partial order, and a group — an algebra equipped with a nullary operation ^[20] that selects a neutral element , a unary operation to obtain the inverse element, and a binary associative operation.

Due to the fact that any $n$ ${\ displaystyle n}$ -ary operation $f\colon A^{n}\to A$ ${\ displaystyle f \ colon A ^ {n} \ to A}$ can be imagined as $(n+1)$ ${\ displaystyle (n + 1)}$ -dimensional relation $\mathrm {r} f=\{\langle a_{1},\;\ldots ,\;a_{n},\;a_{n+1}\rangle \mid a_{n+1}=f(a_{1},\;\ldots ,\;a_{n})\}$ {\ displaystyle \ mathrm {r} f = \ {\ langle a_ {1}, \; \ ldots, \; a_ {n}, \; a_ {n + 1} \ rangle \ mid a_ {n + 1} = f (a_ {1}, \; \ ldots, \; a_ {n}) \}} , any algebraic systems can be investigated as models by model- theoretic tools ^[21] .

Basic Constructions

For algebraic systems, constructions are introduced that are characteristic of all basic general algebraic structures: a subsystem ( subalgebra , submodel ), as a subset of the support of the system, closed with respect to all operations and relations, a homomorphism of systems, as a map between systems of the same type, preserving the basic operations and relations, isomorphism , as a reversible homomorphism, automorphism as an isomorphism onto itself. The introduction of the concept of congruence as a stable equivalence relation on a system allows us to construct such a construction as a factor system ( factor algebra , factor model ) - a system over equivalence classes. Moreover, a homomorphism theorem, general for all algebraic systems, is proved , stating that for any homomorphism $\varphi \colon {\mathfrak {A}}(A,\;\Sigma )\to {\mathfrak {A}}'(A',\;\Sigma )$ ${\ displaystyle \ varphi \ colon {\ mathfrak {A}} (A, \; \ Sigma) \ to {\ mathfrak {A}} '(A', \; \ Sigma)}$ natural mapping of a factor system for nuclear congruence ${\mathfrak {A}}/\{(x,\;y)\in A\times A'\mid \varphi (x)=\varphi (y)\}$ ${\ displaystyle {\ mathfrak {A}} / \ {(x, \; y) \ in A \ times A '\ mid \ varphi (x) = \ varphi (y) \}}$ at ${\mathfrak {A}}'$ ${\ displaystyle {\ mathfrak {A}} '}$ is a homomorphism , and in the case of algebras, an isomorphism .

All subsystems of an algebraic system $\mathbf {Sub} \,{\mathfrak {A}}$ ${\ displaystyle \ mathbf {Sub} \, {\ mathfrak {A}}}$ form a complete lattice , in addition, any algebraic lattice (that is, a lattice, each element of which is represented as the exact upper face of its compact elements) is isomorphic to the lattice of subalgebras of some universal algebra ^[22] . The automorphism groups of algebraic systems are investigated. $\mathbf {Aut} \,{\mathfrak {A}}$ ${\ displaystyle \ mathbf {Aut} \, {\ mathfrak {A}}}$ ^[23] , congruence gratings $\mathbf {Con} \,{\mathfrak {A}}$ ${\ displaystyle \ mathbf {Con} \, {\ mathfrak {A}}}$ . In particular, it is shown that for any group $G$ ${\ displaystyle G}$ and grids $L_{0}$ ${\ displaystyle L_ {0}}$ and $L_{1}$ ${\ displaystyle L_ {1}}$ there is such a universal algebra ${\mathfrak {A}}$ ${\ displaystyle {\ mathfrak {A}}}$ , what $G\cong \mathbf {Aut} \,{\mathfrak {A}}$ ${\ displaystyle G \ cong \ mathbf {Aut} \, {\ mathfrak {A}}}$ , $L_{0}\cong \mathbf {Sub} \,{\mathfrak {A}}$ ${\ displaystyle L_ {0} \ cong \ mathbf {Sub} \, {\ mathfrak {A}}}$ , $L_{1}\cong \mathbf {Con} \,{\mathfrak {A}}$ ${\ displaystyle L_ {1} \ cong \ mathbf {Con} \, {\ mathfrak {A}}}$ .

A direct product is defined over a family of algebraic systems of the same type $\prod _{i\in I}{{\mathfrak {A}}_{i}(A_{i},\;\langle f_{1}\colon A^{n_{1}}\to A,\;\ldots f_{i}\colon A^{n_{i}}\to A,\;\ldots \rangle ,\;\langle r_{1}\subseteq A^{m_{1}},\;\ldots r_{i}\subseteq A^{m_{i}},\;\ldots \rangle )}$ {\ displaystyle \ prod _ {i \ in I} {{\ mathfrak {A}} _ {i} (A_ {i}, \; \ langle f_ {1} \ colon A ^ {n_ {1}} \ to A, \; \ ldots f_ {i} \ colon A ^ {n_ {i}} \ to A, \; \ ldots \ rangle, \; \ langle r_ {1} \ subseteq A ^ {m_ {1}}, \; \ ldots r_ {i} \ subseteq A ^ {m_ {i}}, \; \ ldots \ rangle)}} as a system whose operations and relations are determined coordinate-wise on the Cartesian product of carriers: that is, for $f_{k}\colon (\prod _{i\in I}A_{i})^{n_{k}}\to \prod _{i\in I}A_{i}$ ${\ displaystyle f_ {k} \ colon (\ prod _ {i \ in I} A_ {i}) ^ {n_ {k}} \ to \ prod _ {i \ in I} A_ {i}}$ - $f_{k}(\langle \ldots a_{i,\;1}\ldots \rangle ,\;\ldots ,\;\langle \ldots a_{i,\;n_{k}}\ldots \rangle )=\langle \ldots ,\;f(a_{i,\;1},\;\ldots a_{i,\;n_{k}}),\;\ldots \rangle$ ${\ displaystyle f_ {k} (\ langle \ ldots a_ {i, \; 1} \ ldots \ rangle, \; \ ldots, \; \ langle \ ldots a_ {i, \; n_ {k}} \ ldots \ rangle) = \ langle \ ldots, \; f (a_ {i, \; 1}, \; \ ldots a_ {i, \; n_ {k}}), \; \ ldots \ rangle}$ , and for $r_{k}\subseteq (\prod _{i\in I}A_{i})^{n_{k}}$ ${\ displaystyle r_ {k} \ subseteq (\ prod _ {i \ in I} A_ {i}) ^ {n_ {k}}}$ - $\{\langle \ldots ,\;r(a_{i,\;1},\;\ldots ,\;a_{i,\;k}),\;\ldots \rangle \mid a_{i,\;j}\in A_{i},\;i\in I,\;1\leqslant j\leqslant k\}$ ${\ displaystyle \ {\ langle \ ldots, \; r (a_ {i, \; 1}, \; \ ldots, \; a_ {i, \; k}), \; \ ldots \ rangle \ mid a_ { i, \; j} \ in A_ {i}, \; i \ in I, \; 1 \ leqslant j \ leqslant k \}}$ . The projections of the direct product are natural surjective homomorphisms $\pi _{k}\colon \prod _{i\in I}{{\mathfrak {A}}_{i}}\to {\mathfrak {A}}_{k}$ ${\ displaystyle \ pi _ {k} \ colon \ prod _ {i \ in I} {{\ mathfrak {A}} _ {i}} \ to {\ mathfrak {A}} _ {k}}$ restoring operations and relationships in the components of a work. The Cartesian degree of an algebraic system is a direct product with itself: ${\mathfrak {A}}^{n}=\prod _{i=1}^{n}{{\mathfrak {A}}_{i}}$ ${\ displaystyle {\ mathfrak {A}} ^ {n} = \ prod _ {i = 1} ^ {n} {{\ mathfrak {A}} _ {i}}}$ ; lattice of congruences of algebra $\mathbf {Con} \,{\mathfrak {A}}$ ${\ displaystyle \ mathbf {Con} \, {\ mathfrak {A}}}$ in this sense, we can consider it to be a Cartesian square subalgebra entering the lattice $\mathbf {Sub} \,{\mathfrak {A}}^{2}$ ${\ displaystyle \ mathbf {Sub} \, {\ mathfrak {A}} ^ {2}}$ , moreover, it was established that it is a complete sublattice in it ^[24] .

Varieties

A variety of algebraic systems (or an equational class ) is a class of algebraic systems of a fixed signature, axiomatized by a set of identities expressed in terms of a signature, this concept generalizes such special axiomatically given classes of algebras as the class of all semigroups, the class of all groups, the class of all rings. The basis for the study of such a generalized construction as varieties is the Birkhoff theorem , which states that for axiomatizability by identities of a nonempty class of algebraic systems ${\mathfrak {K}}$ ${\ displaystyle {\ mathfrak {K}}}$ it is necessary and sufficient that it contains:

Cartesian product of an arbitrary sequence ${\mathfrak {K}}$ ${\ displaystyle {\ mathfrak {K}}}$ (was multiplicatively closed );
any subsystem arbitrary ${\mathfrak {K}}$ ${\ displaystyle {\ mathfrak {K}}}$ -systems (was hereditary );
homomorphic image of any ${\mathfrak {K}}$ ${\ displaystyle {\ mathfrak {K}}}$ -systems (was homomorphically closed ) ^[25] .

The third condition is equivalent to being closed with respect to factor systems.

In studies on universal algebra, the structural properties of varieties, the questions of the immersibility of systems of one variety in the systems of another, are studied in detail. In particular, it was established that the lattice of all varieties of lattices is distributive and has a continuum cardinality , and the lattice of all varieties of groups is modular , but not distributive.

In addition to varieties, more general classes of systems have been studied, such as submanifolds ( replica complete classes ) - classes closed with respect to subalgebras and Cartesian products containing a one-element system and quasivarieties - classes axiomatizable instead of a set of identities by a set of quasidentities (defined by Horn clauses ).

Free Algebras

Special Algebras

Categories of Algebraic Systems

Applications

Notes

↑ Whitehead, Alfred North. A treatise on universal algebra, with applications . - Cambridge : Cambridge University Press , 1898. - 547 p.
↑ ¹ ² Cohn, 1969 , p. eleven.
↑ Maltsev, 1970 , p. 7.
↑ Gretzer, 2008 , Although Whitehead recognized the need for universal algebra, he had no results. The first results were published by G. Birkhoff in the thirties, p. vii.
↑ Henkin L. Some interconnections between modern algebra and mathematical logic (Eng.) // Transactions of the American Mathematical Society . - 1953. - Vol. 74 . - P. 410-427 . - ISSN 0002-9947 .
↑ A.I. Maltsev. To the general theory of algebraic systems (rus.) // Mathematical collection . - 1954. - T. 35 , No. 77 . - S. 3-20 .
↑ Abraham Robinson. Note on an embedding theorem for algebraic systems (English) // Journal of the London Mathamtical Society . - 1955. - Vol. 30 . - P. 249-252 .
↑ Bjarni Jónsson. Universal relational systems (English) // Mathematica Scandinavica. - 1957. - No. 5 . - P. 224-229 . - ISSN 0025-5521 .
↑ Maltsev A. I. About one general method for obtaining local theorems of group theory // Uchenye Zapiski Ivanovo State Pedagogical Institute. A series of physical and mathematical sciences. - 1941. - T. 1 , No. 1 . - S. 3-20 .
↑ Gretzer, 2008 , Mal'cev's 1941 paper was the first one, but it went unnoticed because of the war. After the war, A. Tarski, LA Henkin, and A. Robinson began working in this field and they started publishing their results about 1950. A. Tarski's lecture at the International Congress of Mathematicians (Cambridge, Massachusetts, 1950) may be considered as the beginning ofthe new period., p. viii.
↑ Gretzer, 2008 , Marczewski emphasized the importance of bases of free algebras; he called them independent sets. As a result Marczewski, J. Mycielski, W. Narkiewicz, W. Nitka, J. Plonka, S. Swierczkowski, K. Urbanik, and others were responsible for more than 50 papers on the algebraic theory of free algebras, p. viii.
↑ Higgins PJ Groups with multiple operators // Proceedings of the London Mathematical Society. - 1956. - Vol. 6 , no. 3 . - P. 366-416 . - DOI : 10.1112 / plms / s3-6.3.366 .
↑ Kurosh A.G. Lectures on General Algebra - M .: Nauka , 1973. - 400 p.
<a href=" https://wikidata.org/wiki/Track:Q248326 "> </a> <a href=" https://wikidata.org/wiki/Track:Q1384963 "> </a> <a href = " https://wikidata.org/wiki/Track:Q28321667 "> </a>
↑ General Algebra, 1991 , p. 45.
↑ Plotkin B. I. Universal Algebra, Algebraic Logic, and Databases. - M .: Nauka, 1991 .-- 448 p. - 3960 copies. - ISBN 5-02-014635-8 .
↑ Gretzer, 2008 , p. 584.
↑ The Presidium of the RAS decided (October – November 2007) // Bulletin of the Russian Academy of Sciences. - 2008. - T. 78 , no. 3 . - S. 286 .
↑ Maltsev, 1970 .
↑ Gretzer, 2008 , p. 8.
↑ It is assumed that $A^{0}=\varnothing$ ${\ displaystyle A ^ {0} = \ varnothing}$
↑ General Algebra, 1991 , p. 313.
↑ Gretzer, 2008 , Theorem 2, p. 48.
↑ Plotkin B.I. Automorphism groups of algebraic systems. - M .: Nauka , 1966 .-- 603 p. - 6000 copies.
↑ General Algebra, 1991 , p. 302.
↑ Maltsev, 1970 , pp. 337-339.

Literature

Artamonov V.A. Chapter VI. Universal algebras // General algebra / Under the general. ed. L.A. Skornyakova . - M .: Nauka , 1991. - T. 2. - S. 295-367. - 480 p. - (Reference Mathematical Library). - 25,000 copies. - ISBN 5-9221-0400-4 .
Cohn P. Universal Algebra. - M .: Mir , 1969 .-- 351 p.
Maltsev A.I. Algebraic systems. - M .: Nauka , 1970 .-- 392 p. - 17,500 copies.
Grätzer, George. Universal Algebra. - 2nd. - Springer , 2008 .-- 585 p. - ISBN 978-0-387-77486-2 .

[1] Whitehead, Alfred North. A treatise on universal algebra, with applications . - Cambridge : Cambridge University Press , 1898. - 547 p.

[_238c8b7a9a9367e5-2] ¹ ² Cohn, 1969 , p. eleven.

[_be28ea9b7dc39b17-3] Maltsev, 1970 , p. 7.

[_28415f46967140a2-4] Gretzer, 2008 , Although Whitehead recognized the need for universal algebra, he had no results. The first results were published by G. Birkhoff in the thirties, p. vii.

[5] Henkin L. Some interconnections between modern algebra and mathematical logic (Eng.) // Transactions of the American Mathematical Society . - 1953. - Vol. 74 . - P. 410-427 . - ISSN 0002-9947 .

[6] A.I. Maltsev. To the general theory of algebraic systems (rus.) // Mathematical collection . - 1954. - T. 35 , No. 77 . - S. 3-20 .

[7] Abraham Robinson. Note on an embedding theorem for algebraic systems (English) // Journal of the London Mathamtical Society . - 1955. - Vol. 30 . - P. 249-252 .

[8] Bjarni Jónsson. Universal relational systems (English) // Mathematica Scandinavica. - 1957. - No. 5 . - P. 224-229 . - ISSN 0025-5521 .

[9] Maltsev A. I. About one general method for obtaining local theorems of group theory // Uchenye Zapiski Ivanovo State Pedagogical Institute. A series of physical and mathematical sciences. - 1941. - T. 1 , No. 1 . - S. 3-20 .

[_23f0404fa2f6e73d-10] Gretzer, 2008 , Mal'cev's 1941 paper was the first one, but it went unnoticed because of the war. After the war, A. Tarski, LA Henkin, and A. Robinson began working in this field and they started publishing their results about 1950. A. Tarski's lecture at the International Congress of Mathematicians (Cambridge, Massachusetts, 1950) may be considered as the beginning ofthe new period., p. viii.

[_2245b3179e256d2e-11] Gretzer, 2008 , Marczewski emphasized the importance of bases of free algebras; he called them independent sets. As a result Marczewski, J. Mycielski, W. Narkiewicz, W. Nitka, J. Plonka, S. Swierczkowski, K. Urbanik, and others were responsible for more than 50 papers on the algebraic theory of free algebras, p. viii.

[12] Higgins PJ Groups with multiple operators // Proceedings of the London Mathematical Society. - 1956. - Vol. 6 , no. 3 . - P. 366-416 . - DOI : 10.1112 / plms / s3-6.3.366 .

[13] Kurosh A.G. Lectures on General Algebra - M .: Nauka , 1973. - 400 p.
<a href=" https://wikidata.org/wiki/Track:Q248326 "> </a> <a href=" https://wikidata.org/wiki/Track:Q1384963 "> </a> <a href = " https://wikidata.org/wiki/Track:Q28321667 "> </a>

[_3e6f72c10ff7b0b6-14] General Algebra, 1991 , p. 45.

[15] Plotkin B. I. Universal Algebra, Algebraic Logic, and Databases. - M .: Nauka, 1991 .-- 448 p. - 3960 copies. - ISBN 5-02-014635-8 .

[_1be87b00d92e42c3-16] Gretzer, 2008 , p. 584.

[17] The Presidium of the RAS decided (October – November 2007) // Bulletin of the Russian Academy of Sciences. - 2008. - T. 78 , no. 3 . - S. 286 .

[_6d4c81ca37052860-18] Maltsev, 1970 .

[_89ac7f469cd1d59a-19] Gretzer, 2008 , p. 8.

[20] It is assumed that $A^{0}=\varnothing$ ${\ displaystyle A ^ {0} = \ varnothing}$

[_0f21b00e21efb19a-21] General Algebra, 1991 , p. 313.

[_237aa219a93900a4-22] Gretzer, 2008 , Theorem 2, p. 48.

[23] Plotkin B.I. Automorphism groups of algebraic systems. - M .: Nauka , 1966 .-- 603 p. - 6000 copies.

[_0f21af0e21efafc4-24] General Algebra, 1991 , p. 302.

[_13ba9df31626eadc-25] Maltsev, 1970 , pp. 337-339.