Editing Universal algebra

{{Short description|Theory of algebraic structures in general}}
'''Universal algebra''' (sometimes called '''general algebra''') is the field of [[mathematics]] that studies [[algebraic structure]]s themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular [[Group (mathematics)|groups]] as the object of study, in universal algebra one takes the [[class of groups]] as an object of study.

== Basic idea ==
{{Main|Algebraic structure}}
{{distinguish|Algebra over a field}}
In universal algebra, an '''{{vanchor|algebra}}''' (or [[algebraic structure]]) is a [[set (mathematics)|set]] ''A'' together with a collection of operations on ''A''.  

=== Arity ===
{{Main|Arity}}
An '''''n''-[[arity|ary]] [[operation (mathematics)|operation]]''' on ''A'' is a [[function (mathematics)|function]] that takes ''n'' elements of ''A'' and returns a single element of ''A''.  Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a ''[[Constant (mathematics)|constant]]'', often denoted by a letter like ''a''.  A 1-ary operation (or ''[[unary operation]]'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''.  A 2-ary operation (or ''[[binary operation]]'') is often denoted by a symbol placed between its arguments (also called [[infix notation]]), like ''x''&nbsp;∗&nbsp;''y''.  Operations of higher or unspecified ''[[arity]]'' are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like ''f''(''x'',''y'',''z'') or ''f''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>).   One way of talking about an algebra, then, is by referring to it as an [[Outline of algebraic structures#Types of algebraic structures|algebra of a certain type]] <math>\Omega</math>, where <math>\Omega</math> is an ordered sequence of natural numbers representing the arity of the operations of the algebra.  However, some researchers also allow [[infinitary]] operations, such as <math>\textstyle\bigwedge_{\alpha\in J} x_\alpha</math> where ''J'' is an infinite [[index set]], which is an operation in the algebraic theory of [[complete lattice]]s. 

=== Equations ===
After the operations have been specified, the nature of the algebra is further defined by [[axiom]]s, which in universal algebra often take the form of [[Identity (mathematics)#Logic and universal algebra|identities]], or '''equational laws.'''    An example is the [[associative]] axiom for a binary operation, which is given by the equation ''x''&nbsp;∗&nbsp;(''y''&nbsp;∗&nbsp;''z'')&nbsp;= (''x''&nbsp;∗&nbsp;''y'')&nbsp;∗&nbsp;''z''.  The axiom is intended to hold for all elements ''x'', ''y'', and ''z'' of the set ''A''.

== Varieties ==
{{Main|Variety (universal algebra)}}

A collection of algebraic structures defined by identities is called a [[Variety (universal algebra)|variety]] or '''equational class'''.

Restricting one's study to varieties rules out:
* [[Quantification (logic)|quantification]], including  [[universal quantification]] (∀) except before an equation, and [[existential quantification]] (∃)
* [[logical connective]]s other than [[logical conjunction|conjunction]] (∧)
* [[Finitary relation|relations]] other than equality, in particular [[inequality (mathematics)|inequalities]], both {{nowrap|''a'' ≠ ''b''}} and [[Order theory|order relations]]

The study of equational classes can be seen as a special branch of [[model theory]], typically dealing with structures having operations only (i.e. the [[signature (logic)|type]] can have symbols for functions but not for [[Finitary relation|relations]] other than equality), and in which the language used to talk about these structures uses equations only.

Not all [[algebraic structure]]s in a wider sense fall into this scope. For example, [[ordered group]]s involve an ordering relation, so would not fall within this scope.

The class of [[field (mathematics)|field]]s is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all ''non-zero'' elements in a field, so inversion cannot be added to the type).

One advantage of this restriction is that the structures studied in universal algebra can be defined in any [[category theory|category]] that has ''finite [[product (category theory)|product]]s''. For example, a [[topological group]] is just a group in the category of [[topological space]]s.

=== Examples ===
Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.

==== Groups ====
As an example, consider the definition of a [[group (mathematics)|group]].  Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
* [[associative|Associativity]] (as in the [[#Equations|previous section]]):  ''x''&nbsp;∗&nbsp;(''y''&nbsp;∗&nbsp;''z'')&nbsp;&nbsp;=&nbsp; (''x''&nbsp;∗&nbsp;''y'')&nbsp;∗&nbsp;''z''; &nbsp; formally: ∀''x'',''y'',''z''. ''x''∗(''y''∗''z'')=(''x''∗''y'')∗''z''.
* [[Identity element]]:  There exists an element ''e'' such that for each element ''x'', one has ''e''&nbsp;∗&nbsp;''x''&nbsp;&nbsp;=&nbsp; ''x''&nbsp;&nbsp;=&nbsp; ''x''&nbsp;∗&nbsp;''e''; &nbsp; formally: ∃''e'' ∀''x''. ''e''∗''x''=''x''=''x''∗''e''.
* [[Inverse element]]:  The identity element is easily seen to be unique, and is usually denoted by ''e''. Then for each ''x'', there exists an element ''i'' such that ''x''&nbsp;∗&nbsp;''i''&nbsp;&nbsp;=&nbsp; ''e''&nbsp;&nbsp;=&nbsp; ''i''&nbsp;∗&nbsp;''x''; &nbsp; formally: ∀''x''&nbsp;∃''i''.&nbsp;''x''∗''i''=''e''=''i''∗''x''.

(Some authors also use the "[[Closure (mathematics)|closure]]" axiom that ''x''&nbsp;∗&nbsp;''y'' belongs to ''A'' whenever ''x'' and ''y'' do, but here this is already implied by calling ∗ a binary operation.)

This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all&nbsp;..." elements, but also involve the existential quantifier "there exists&nbsp;...".  The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation ''e'' and a unary operation ~, with ~''x'' usually written as ''x''<sup>−1</sup>. The axioms become:
* Associativity:  {{nowrap|1=''x'' ∗ (''y'' ∗ ''z'') &nbsp;=&nbsp;}} {{nowrap|(''x'' ∗ ''y'') ∗ ''z''}}.
* Identity element:  {{nowrap|1=''e'' ∗ ''x'' &nbsp;=&nbsp;}} {{nowrap|1=''x'' &nbsp;=&nbsp;}} {{nowrap|''x'' ∗ ''e''}}; &nbsp; formally: {{nowrap|1=∀''x''. ''e''∗''x''=''x''=''x''∗''e''}}.
* Inverse element:  {{nowrap|1=''x'' ∗ (~''x'') &nbsp;=&nbsp;}} {{nowrap|1=''e'' &nbsp;=&nbsp;}} {{nowrap|(~''x'') ∗ ''x''}}; &nbsp; formally: {{nowrap|1=∀''x''. ''x''∗~''x''=''e''=~''x''∗''x''}}.

To summarize, the usual definition has:
* a single binary operation ([[signature (logic)|signature]] (2))
* 1 equational law (associativity)
* 2 quantified laws (identity and inverse)
while the universal algebra definition has:
* 3 operations: one binary, one unary, and one nullary ([[signature (logic)|signature]] {{nowrap|(2, 1, 0)}})
* 3 equational laws (associativity, identity, and inverse)
* no quantified laws (except outermost universal quantifiers, which are allowed in varieties)

A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element ''e'', an easy exercise shows that it is unique, as is the [[inverse element|inverse]] of each element.

The universal algebra point of view is well adapted to category theory. For example, when defining a [[group object]] in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements). Further, the inverse and identity are specified as morphisms in the category. For example, in a [[topological group]], the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a [[closed inclusion]] (a [[cofibration]]).

==== Other examples ====
Most algebraic structures are examples of universal algebras.
* [[Ring (mathematics)|Rings]], [[semigroup]]s, [[quasigroup]]s, [[groupoid]]s, [[Magma (mathematics)|magmas]], [[Loop (algebra)|loops]], and others.
* [[Vector space]]s over a fixed field and [[module (mathematics)|modules]] over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring.

Examples of relational algebras include [[semilattice]]s, [[lattice (order)|lattices]], and [[Boolean algebra]]s.

== Basic constructions ==
We assume that the type, <math>\Omega</math>, has been fixed.  Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

A [[homomorphism]] between two algebras ''A'' and ''B'' is a [[function (mathematics)|function]] {{nowrap|''h'' : ''A'' → ''B''}} from the set ''A'' to the set ''B'' such that, for every operation ''f''<sub>''A''</sub> of ''A'' and corresponding ''f''<sub>''B''</sub> of ''B'' (of arity, say, ''n''), ''h''(''f''<sub>''A''</sub>(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>))&nbsp;= ''f''<sub>''B''</sub>(''h''(''x''<sub>1</sub>), ..., ''h''(''x''<sub>''n''</sub>)).  (Sometimes the subscripts on ''f'' are taken off when it is clear from context which algebra the function is from.) For example, if ''e'' is a constant (nullary operation), then ''h''(''e''<sub>''A''</sub>)&nbsp;=&nbsp;''e''<sub>''B''</sub>.  If ~ is a unary operation, then ''h''(~''x'')&nbsp;= ~''h''(''x'').  If ∗ is a binary operation, then ''h''(''x''&nbsp;∗&nbsp;''y'')&nbsp;= ''h''(''x'')&nbsp;∗&nbsp;''h''(''y'').  And so on.  A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under ''[[Homomorphism]]''.  In particular, we can take the homomorphic image of an algebra, ''h''(''A'').

A subalgebra of ''A'' is a subset of ''A'' that is closed under all the operations of ''A''.  A product of some set of algebraic structures is the [[cartesian product]] of the sets with the operations defined coordinatewise.

== Some basic theorems ==
* The [[isomorphism theorems]], which encompass the isomorphism theorems of [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], [[Module (mathematics)|modules]], etc.
* [[Variety (universal algebra)#Birkhoff's theorem|Birkhoff's HSP Theorem]], which states that a class of algebras is a [[variety (universal algebra)|variety]] if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.

== Motivations and applications ==
{{Unreferenced section|date=April 2010}}

In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study of new classes of algebras.
It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it, ''"What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."''

In particular, universal algebra can be applied to the study of [[monoid]]s, [[ring (algebra)|rings]], and [[lattice (order)|lattice]]s.  Before universal algebra came along, many theorems (most notably the [[isomorphism theorem]]s) were proved separately in all of these classes, but with universal algebra, they can be proven once and for all for every kind of algebraic system.

The 1956 paper by Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially defined, typical examples for this being categories and groupoids. This leads on to the subject of [[higher-dimensional algebra]] which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.

=== Constraint satisfaction problem ===
{{Main|Constraint satisfaction problem}}

Universal algebra provides a natural language for the [[constraint satisfaction problem|constraint satisfaction problem (CSP)]]. CSP refers to an important class of computational problems where, given a relational algebra ''A'' and an existential [[sentence (mathematical logic)|sentence]] <math>\varphi</math> over this algebra, the question is to find out whether <math>\varphi</math> can be satisfied in ''A''. The algebra ''A'' is often fixed, so that CSP<sub>''A''</sub> refers to the problem whose instance is only the existential sentence <math>\varphi</math>.

It is proved that every computational problem can be formulated as CSP<sub>''A''</sub> for some algebra ''A''.<ref>{{Citation|last1=Bodirsky|first1=Manuel|last2=Grohe|first2=Martin|date=2008|title=Non-dichotomies in constraint satisfaction complexity|url=http://www.lix.polytechnique.fr/~bodirsky/publications/nodich.pdf}}</ref>

For example, the [[graph coloring|''n''-coloring]] problem can be stated as CSP of the algebra {{nowrap|({{mset|0, 1, ..., ''n''−1}}, ≠)}}, i.e. an algebra with ''n'' elements and a single relation, inequality.

== Generalizations ==
{{Further|Category theory|Operad theory|Partial algebra|Model theory}}

Universal algebra has also been studied using the techniques of [[category theory]].  In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as [[Lawvere theory|Lawvere theories]] or more generally [[algebraic theory|algebraic theories]].  Alternatively, one can describe algebraic structures using [[monad (category theory)|monad]]s.  The two approaches are closely related, with each having their own advantages.<ref>
{{Citation|last1=Hyland|first1=Martin|last2=Power|first2=John|date=2007|title=The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads|url =http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf |archive-url=https://web.archive.org/web/20230530165901/http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf |archive-date=30 May 2023}}</ref>
In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory.  However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure within any of a large class of categories (namely those having finite [[product (category theory)|products]]).

A more recent development in category theory is [[operad theory]]&nbsp;– an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed.<ref>{{Cite book |author1=Markl, M. |title=Operads in Algebra, Topology and Physics |author2=Shnider, S. |author3=Stasheff, J. D. |publisher=American Mathematical Society |year=2002 |isbn=9780821843628 |series=Mathematical Surveys and Monographs |lccn=2002016342}}</ref> Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law {{nowrap|1=''gg''<sup>−1</sup> = 1}} duplicates the variable ''g'' on the left side and omits it on the right side.  At first this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of ring and vector space to obtain the concept of [[associative algebra]], but one cannot form a similar hybrid of the concepts of group and vector space.<ref>{{Cite book |author=Pierce, Richard S. |title=Associative Algebras |publisher=Springer New York |year=1982 |isbn=978-1-4757-0163-0 |location=New York, NY |pages=1–20 |chapter=The Associative Algebra |series=Graduate Texts in Mathematics |volume=88 |doi=10.1007/978-1-4757-0163-0_1}}</ref>

Another development is [[partial algebra]] where the operators can be [[partial function]]s.  Certain partial functions can also be handled by a generalization of Lawvere theories known as "essentially algebraic theories".<ref>{{nlab|id=essentially+algebraic+theory|title=Essentially algebraic theory}}</ref>

Another generalization of universal algebra is [[model theory]], which is sometimes described as "universal algebra + logic".<ref>{{cite book | isbn=0444880542 | author=C.C. Chang and H. Jerome Keisler | title=Model Theory | publisher=North Holland | series=Studies in Logic and the Foundation of Mathematics | volume=73 | edition=3rd | year=1990 |page=1}}</ref>

== History ==
In [[Alfred North Whitehead]]'s book ''A Treatise on Universal Algebra,'' published in 1898, the term ''universal algebra'' had essentially the same meaning that it has today. Whitehead credits [[William Rowan Hamilton]] and [[Augustus De Morgan]] as originators of the subject matter, and [[James Joseph Sylvester]] with coining the term itself.<ref name="Gratzer.1968">{{cite book | author=George Grätzer|editor=M.H. Stone and L. Nirenberg and S.S. Chern |title=Universal Algebra|publisher=Van Nostrand Co., Inc|edition=1st| year=1968 }}</ref>{{rp|v}}

At the time structures such as [[Lie algebra]]s and [[hyperbolic quaternion]]s drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review [[Alexander Macfarlane]] wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures."<ref>[[Alexander Macfarlane]] (1899) [https://archive.org/details/jstor-1626993 Review:''A Treatise on Universal Algebra'' (pdf)], [[Science (journal)|Science]] 9: 324–8 via [[Internet Archive]]</ref> At the time [[George Boole]]'s algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.

Whitehead's early work sought to unify [[quaternions]] (due to Hamilton), [[Grassmann]]'s [[Exterior algebra#History|Ausdehnungslehre]], and Boole's algebra of logic.  Whitehead wrote in his book:
:''"Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge."''<ref name="Gratzer.1968"/>

Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when [[Garrett Birkhoff]] and [[Øystein Ore]] began publishing on universal algebras.  Developments in [[metamathematics]] and [[category theory]] in the 1940s and 1950s furthered the field, particularly the work of [[Abraham Robinson]], [[Alfred Tarski]], [[Andrzej Mostowski]], and their students.<ref>Brainerd, Barron (Aug–Sep 1967) "Review of ''Universal Algebra'' by [[P. M. Cohn]]", [[American Mathematical Monthly]] 74(7): 878–880.</ref>

In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers, dealing with [[Free object|free algebras]], congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by [[Anatoly Maltsev]] in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950 [[International Congress of Mathematicians]] in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, [[Leon Henkin]], [[Bjarni Jónsson]], [[Roger Lyndon]], and others.

In the late 1950s, [[Edward Marczewski]]<ref>Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. '''6''' (1958), 731–736.</ref> emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with [[Jan Mycielski]], Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, [[Kazimierz Urbanik|K. Urbanik]], and others.

Starting with [[William Lawvere]]'s thesis in 1963, techniques from category theory have become important in universal algebra.<ref>{{Citation|last1=Lawvere|first1=William F.|author-link=William Lawvere|date=1964|title=Functorial Semantics of Algebraic Theories (PhD Thesis)|url = http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html}}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Equational logic]]
* [[Graph algebra]]
* [[Term algebra]]
* [[Clone (algebra)|Clone]]
* [[Universal algebraic geometry]]
* [[Simple algebra (universal algebra)]]

== Footnotes ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation | last1=Bergman | first1=George M. | year=1998 | url=http://math.berkeley.edu/~gbergman/245/ | title=An Invitation to General Algebra and Universal Constructions | publisher=Henry Helson | location=Berkeley CA |  page=398 | isbn=0-9655211-4-1 }}.
* {{citation | last1=Birkhoff | first1=Garrett | year=1946 | title=Universal algebra | journal=Comptes Rendus du Premier Congrès Canadien de Mathématiques | publisher=University of Toronto Press | location=Toronto | pages = 310–326 }}
* Burris, Stanley N., and H.P. Sankappanavar, 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra]''  Springer-Verlag. {{isbn|3-540-90578-2}} ''Free online edition''.
* {{citation | last1=Cohn | first1=Paul Moritz | year=1981 | title=Universal Algebra | location=Dordrecht, Netherlands | publisher=D. Reidel Publishing | isbn=90-277-1213-1 }} (First published in 1965 by Harper & Row)
* Freese, Ralph, and Ralph McKenzie, 1987. ''[http://www.math.hawaii.edu/~ralph/Commutator Commutator Theory for Congruence Modular Varieties], 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. {{isbn|0-521-34832-3}}. Free online second edition''.
* {{citation | last1=Grätzer | first1=George | year=1968 | title=Universal Algebra | publisher=D. Van Nostrand Company, Inc. }}
* {{citation | last1=Higgins | first1=P. J. | url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-6.3.366 | title=Groups with multiple operators | journal=Proc. London Math. Soc. | volume=3 | issue=6 | year=1956 | pages=366–416 | doi=10.1112/plms/s3-6.3.366 }}
* {{citation | last1=Higgins | first1=P. J. | title=Algebras with a scheme of operators | journal=[[Mathematische Nachrichten]] | volume=27 | year=1963 | issue=1–2 | pages=115–132 | doi=10.1002/mana.19630270108 }}
* Hobby, David, and Ralph McKenzie, 1988. ''[http://www.ams.org/books/conm/076/ The Structure of Finite Algebras]'' American Mathematical Society. {{isbn|0-8218-3400-2}}. ''Free online edition.''
* Jipsen, Peter, and Henry Rose, 1992. ''[http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html Varieties of Lattices]'', Lecture Notes in Mathematics 1533. Springer Verlag. {{isbn|0-387-56314-8}}. ''Free online edition''.
* Pigozzi, Don. ''[http://people.math.sc.edu/mcnulty/alglatvar/pigozzinotes.pdf General Theory of Algebras]''.  ''Free online edition.''
* {{citation | last1=Smith | first1=J.D.H. | year=1976 | title=Mal'cev Varieties | publisher=Springer-Verlag }}
* {{citation | last1=Whitehead | first1=Alfred North | author-link1=Alfred North Whitehead | year=1898 | url=https://babel.hathitrust.org/cgi/pt?id=coo.31924059413124&seq=5 | title=A Treatise on Universal Algebra | publisher=Cambridge }}  (''Mainly of historical interest.'')
{{refend}}

== External links ==
* [https://www.springer.com/birkhauser/mathematics/journal/12 ''Algebra Universalis'']—a journal dedicated to Universal Algebra.

{{Algebra}}
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[[Category:Universal algebra| ]]