Universal algebra - SpeedyLook encyclopedia

The universal algebra is the branch of the algebra the purpose of which is to treat in a general and simultaneous way the various algebraic structures: group S, Monoid S, ring X, vector spaces, etc It makes it possible to define in a uniform way homomorphisms, substructures (vectorial sub-groups, under-monoids, subrings, subspaces, etc), the quotients, the products and the objects free for these structures.

There is in mathematics a great number of the algebraic types of structures which each one checks various axioms (groups, rings, vector spaces, treilli S, Boolean algebra, algebras of Dregs). There is, for these various types of structures, a concept of Homomorphisme S and constructions of structures which are similar or which have similar properties (substructures, quotients, produced, free coproduits, objects, limits projective and inductive, etc), and these homomorphisms and these constructions have a great number of properties which are similar (the intersection of sub-groups, subrings, etc, in is one, the image of a sub-group, a subring, etc, by a homomorphism, also), and which is often taught in the first years of the university in mathematics, at least for groups, rings and vector spaces. One then defined in a way general and abstracted the algebraic structures to be able to treat in a uniform way these constructions and their properties, and one could, thereafter, concentrate on the properties suitable for each one of his structures.

More than one generalization of the usual algebraic structures which would be useful only in algebra, the universal algebra has also applications in logic and data processing.

Algebras

Preliminary example

It can be useful to examine an example to release a unifying concept of algebraic Structure. Sometimes, in the algebraic definition of structure one limits oneself to the data of laws of composition interns and external on a unit, but the data of these laws always does not make it possible to define homomorphisms as the applications which preserve these laws and the substructures (Sous-groupe S, Sous-anneau X, etc) like stable parts for laws. Here is an example:

That is to say $G$ a group, φ its law of composition, $e$ its neutral element, γ the application which with an element of $G$ associates its Inverse. That is to say $S$ part of $G$ . It is not enough that $S$ contains it made up of two unspecified elements of $S$ so that $S$ is a Sous-groupe of $G$ , as the case of the whole of the natural entireties shows it $\ mathbb {NR}$ in the group $\ mathbb {Z}$ of the rational entireties. In fact, so that $S$ is a sub-group of $G$ , it is necessary and it is also enough that it contains the neutral element and that he contains the reverse of each one of his elements. In other words, $S$ is a sub-group of $G$ if and only if, φ ( S × S ) ⊂ S , γ ( S ) ⊂ S and E ∈ S. Ainsi, the structure of group of $G$ is not complete without γ and $e$ . There are similar results for the monoids, the rings, the vector spaces, the Boolean algebra, etc

If it is wanted that the algebraic structure determines the substructures (left stable by the various laws), one must enrich the usual structures (groups, rings, etc) of additional laws.

Algebras

] The S usual are not only determined by laws, but by axioms (associativeness, commutation, distributivity, neutral element, etc) which govern the laws: identities. One will see later a general standard of identities. Then let us note that so algebraic structures vérififient all certain common identities, the majority of the usual constructions (substructures, quotients, products) deduced from these structures checks them. One concentrates initially on the contructions of structures defined by laws which do not check any particular identity, and the stability of the identities by these constructions will make so that the results so obtained will apply word for word to the structures defined by the data of laws checking certain identities.

A algebra is a unit provided with a algebraic Structure, whose one will give a precise definition. Among the algebras, one finds the group S, the Monoïde S, the Module S on a given ring (in particular the vector spaces on a given body), the lattice and the Boolean algebra.

Definition. Is $E$ a Ensemble and $n$ a Entier naturalness (no one or not). One calls operation $n$ -aire on $E$ any application of $E^n$ in $E$ . If $n = 2$ , then it acts of a Law of composition intern on $E$ . If $n = 1$ , it acts of an application of E in E , and they are called unary operations in $E$ . If there is $n = 0$ (one speaks then about operations nullaires on $E$ ), they are identified with the elements of $E$ , since $E^0 = \ {0 \}$ and that the applications of {0} in E are identified with the single element of their image. One calls operation finitaire on $E$ an application which is an operation p - surface on $E$ for a natural entirety p .

That is to say $A$ a unit. Then a external Law of composition or a external law of $A$ on $E$ is, by definition, an application of has × E in $E$ . An external law of composition of has on E is identified with an application of has the unit $E^ {E}$ of the applications of E in E , therefore unary operations of E : with an element has , one associates the partial application definite by has this external law. Thus the whole of the external laws of composition of has on E is identified with the whole of the applications of has in $E^ {E}$ , and thus to the whole of the unary families of operations of $E$ indexed by $A$ . Let us note that the distinction between the external laws of composition on the left and on the right is more one business of notation that a major distinction, since there is a canonical bijection between the two.

Definition. a universal algebra or more simply a algebra (not to confuse with the algebra S on a commutative ring, that one meets in Linear algebra) is a unit has provided with a family (vacuum or not) of operations finitaires on has , and one says that the unit has , is subjacent with the algebra in question. A N - uplet (for example a couple) of operations finitaires is a family of operations finitaires.

Certain authors suppose that algebra is not empty, but that is useless.

For allow simultaneously to treat several algebras, it can be useful to determine if they are of the same signature, i.e. if they all are of the rings or all of the lattices, for examples. For that, one owes paramétriser the operations finitaires of the algebras by certain units, the operations N - of the same surfaces parameter corresponding. For example, usually, in the data of the laws of a ring, the addition comes before the multiplication, and thus it is known that the addition of a ring corresponds to the addition of another ring, and not with its multiplication.

Definition. Let us give a unit Ω (vacuum or not) provided, for entire naturalness N of a part (vacuum or not) of Ω, which one notes $\ Omega_ {N}$ . It is said whereas one gave oneself a signature or a standard of algebra or a field of operators . A algebra or a universal algebra of signature Ω is a unit has provided, for entire naturalness N and any element ω of $\ Omega_ {N}$ of an operation N - surface on has , that one will note $\ omega_A$ , or ω if no confusion results from it. It is said whereas the data of these operations finitaires defines on the unit a algebraic Structure of signature Ω.

Convention . In what follows, one is given once for a whole signature of algebra and all the algebras are supposed to be of this signature, except contrary mention.

If there is an algebra has , one calls constant or element distinguished from has the elements of has to which are identified the operations nullaires definite by the elements of $\ Omega_0$ .

Examples of algebras

Let us give some examples of algebras.

a Ensemble is an algebra provided with any operation finitaire (it is not excluded that the signature of algebra is empty).
a Ensemble pointed is a unit E provided with an element of E , i.e. of an operation nullaire.
a magma is a unit provided with a law of composition, i.e. of a binary operation.
a Monoïde is a magma provided with a neutral element. It is thus provided with a binary operation and an operation nullaire.
a group is a monoid such as any element admits a reverse and it is thus provided with the unary operation which with an element associates its reverse. Thus it is provided with a binary operation, an operation nullaire and a unary operation.
a ring is at the same time a group for the addition and a monoid for the multiplication. It is thus provided with two binary operations, two operations nullaires (0 and 1) and with a unary operation.
a Module on a ring has is a commutative group provided with an external law of has checking certain properties. It is thus provided with a binary operation (addition), an operation nullaire (0), of a unary operation (which with X associates - X ) and, for any element of has of a unary operation (corresponding homothety).
a lattice is a unit provided with two laws of composition checking certain properties (limits higher and limits lower).
a Boolean algebra is a lattice and is provided with sound smaller element, sound larger element and the application which with an element associates its complementary. Thus it is provided with two binary operations, two operations nullaires (0 and 1) and with a unary operation.

Like shows it these examples, to describe the majority of the usual algebraic structures, one could limit oneself to the algebras which have only binary operations, operations nullaires and unary operations. But, even for these structures, it can be useful to consider the unspecified operations finitaires, since, with an associative law of composition, one can, for entire naturalness not no one N , to define an operation N - surface which with N elements associates their product (or their nap).

In the usual definitions of these structures, the unit is not provided with all these operations finitaires, but that some of enters, and the axioms of these structures imply the existence of these others.

Let us note that these examples allow of déterminier all the structure which is necessary to correctly define homomorphisms and the substructures (Sous-groupe S, Sous-anneau X, etc).

The body are rings nonreduced to an element of which any element not no one is invertible. In a body, the application which with an element not no one associates its reverse is not defined (0 are not invertible), and thus is not a unary operation on the entire body, but only if the group of its invertible elements. To mitigate this inconvéniant, one must work out the theory of the partial algebras , where the operations finitaires are not defined everywhere.

Here two commonplace examples, which have their importance.

For any unit E which is a singleton, there exists a single algebraic structure of signature Ω on E . An algebra whose subjacent whole is a singleton is known as commonplace .
There exists on the Empty set with more the one algebraic structure of signature Ω, and so that there is one, it is necessary and there is enough that the unit $\ Omega_0$ of the parameters of the operations nullaires is empty, it is to say that there is no operation nullaire. For example, there does not exist empty group, but there exists single a magma vacuum and there exists a single empty lattice.

Let us note that the definition of algebra which one gives makes so that one admits the ring X commonplace and the Boolean algebra commonplace, i.e. reduced to an element. Certain authors exclude these rings and these Boolean algebra.

Homomorphisms

One gives here a general standard of the Homomorphisme S which includes homomorphisms of group S, of ring X, Module S (linear applications), etc

Definition. Is has and B of the same algebras signature Ω. a Homomorphisme (or Ω- homomorphism or a homomorphism of Ω-algebras if it should be specified) has in B , or, if has = B , a endomorphism of has , is a application of has in B which preserves the operations finitaires corresponding to the same elements of Ω. In a more precise way, an application F of has in B is a homomorphism if and only if, for entire naturalness N and for any element ω of $\ Omega_n$ , $\ omega_B$ ( F ( $x_1$ ),…, F ( $x_n$ )) = F ( $\ omega_A$ ( $x_1$ ,…, $x_n$ )), whatever the elements $x_1$ ,…, $x_n$ of has. Homomorphisms bear also the name of Morphisme

It is seen here that homomorphisms such as here definite correspond to homomorphisms of the various algebraic structures which one meets in general algebra (Monoïde S, groups, rings, modules on a given ring, treilli S, etc).

Let us note that it may be that, for certain algebras has and B , there do not exist homomorphisms of has in B , even if has and B is not vacuums.

Proposition Is has , B and C of the algebras. The made up one of a homomorphism of has in B and of a homomorphism of B in C is a homomorphism of has in C . The identity application of an algebra has is a endomorphism of has . In terms of Theory of the categories, the class of the algebras of signature Ω with homomorphisms form a category for the composition of homomorphisms (as applications).

Definition. Is has and B of the same algebras signature. One calls Isomorphisme of has on B or, if has = B , Automorphisme of has , any homomorphism of has in B which is bijective. For any isomorphism F of has on B , the reciprocal bijection of F is an isomorphism of B on has . That coincides with the concept of isomorphism in theory of the categories.

Proposal. the whole of the endomorphisms of has is a monoid for the composition of homomorphisms, which one notes End ( has ). The whole of the automorphisms of has is a group for the composition of homomorphisms, which one notes Aut ( has ). It is the group of the invertible elements of the End monoid ( has ).

If they exist, then the empty algebras are the only initial objects of the category of the algebras of signature Ω, i.e., for any algebra has , it exists a homorphism of the Empty set in has . If there does not exist of empty algebra, there are also initial objects, but it are not empty. The commonplace algebras are the only final objects of this category, i.e., for any algebra has , it exists a single homomorphism of has in a commonplace algebra.

Subalgebras

Definition of the subalgebras

One defines here the concepts of subalgebra, which generalizes the usual concepts of induced structures (or substructures) of the usual algebraic structures, for examples the Sous-groupe S, the Sous-anneau X, the Sous-module S (or vectorial subspaces), etc

One is given once for all an algebra has signature Ω.

Definition. a subalgebra of has (or under - Ω- algebra of has if one makes a point of specifying) is a part of has which is stable for each operation finitaires of has . In a more precise way, a part S of has is a subalgebra of has if and only if, for entire naturalness N , any element ω of $\ Omega_n$ and whatever the elements $x_1$ ,…, $x_n$ of S , $\ omega_A$ ( $x_1$ ,…, $x_n$ ) belongs to S . If N = 0, that means that the element $\ omega_A$ of has belongs to S .

If S is a subalgebra of has , then by restriction on S of the operations of has , one obtains an algebraic structure of signature Ω on the unit S , known as induced by that of has , and thus S is thus canonically an algebra of signature Ω. When one regards S as an algebra, it is for this algebraic structure on S .

One can check that the subalgebras correspond well to the induced structures for the usual algebraic structures: subsets pointed, under-magmas, Under-monoid S, Sub-group S, Subring X, Submodule S (or vectorial subspaces), subalgebra of an algebra on a commutative ring, unit subalgebras of a unit algebra on a commutative ring, under-lattices, subalgebras of Boole, etc However, the subalgebras of the bodies are its subrings, and not necessarily all its subfields (the inversion is not regarded here as a unary operation, but not everywhere definite).

Properties of the subalgebras

Here elementary properties of the subalgebras. The reader will be can be familiar with the similar statements in the cases of the groups, the rings or the modules. One indicates by has an algebra.

has is a subalgebra of has and, if it $\ omega_0$ is empty (thus if there not of elements distinguished from is ), then the empty set is a subalgebra of has .
For any subalgebra B of has , the canonical injection of B in has is a homomorphism.
the intersection of a whole of subalgebras of has is a subalgebra of has .
Any subalgebra of a subalgebra of has is a subalgebra of has (heridity of the subalgebras).
Is B an algebra, F a homomorphism of has in B . Then the image by F of a subalgebra of has is a subalgebra of B and the reciprocal image by F of a subalgebra of B is a subalgebra of has . In particular, the image of F is a subalgebra of B .
Is B an algebra, F a homomorphism of has in B , C and D of the subalgebras of has and B respectively. If F ( C ) is included in D , then the application of C in D which coincides with F is a homomorphism. In particular, the restriction of F on C is a homomorphism of C in B .
the whole of the fixed points of a endomorphism of has is a subalgebra of has .
Is B algebras, F and G of homomorphisms of has in B . Then the whole of the elements X of has such as F ( X ) = G ( X ) is a subalgebra of has .
the meeting of a unit E of subalgebras of has which is filtant for the relation of inclusion (i.e., if has and B belongs to E , then there exists an element of E which contains has and B ) is a subalgebra of has , called, filter meeting of these subalgebras. In particular, the meeting of a completely ordered whole of subalgebras of has is a subalgebra of has .

Let us note that the meeting of subalgebras is not always a subalgebra. For example, the two vectorial line meeting distinct from a vector space E on a body is not a vectorial subspace of E .

The authors who exclude the empty algebras, generally exclude the empty subalgebras, and then the intersection of subalgebras is not necessarily a subalgebra, except if it is nonempty.

Generated subalgebras

Definition. Is X part of an algebra has . The intersection of the whole of the subalgebras of has which contain X ( has is one) is a subalgebra G of has , which is known as generated by X . (That generalizes the generated sub-groups generated, subrings, the generated submodules, etc) If G = has , then one says that X is a generating part of has and that X generates has . One defines in a similar way the generating families and the families which generate .

Definition. If there exists a finished generating part of an algebra has , one says that has is of the type finished . If there exists an element of has which generates has , then one says that has is monogene . That generalizes the similar concepts which one meets in theory of the groups, the rings, of the modules, etc

Proposal. For the relation of inclusion, the whole of the subalgebras of an algebra has is a complete lattice, i.e., any whole of subalgebras admits a lower limit and an upper limit for the relation of inclusion. The limits lower of a family of subalgebras is their intersection and the limits higher the subalgebra generated by their meeting.

Here some properties of the generated subalgebras.

Is has an algebra. The application which with very part X of has associates the subalgebra of has generated by X is increasing, for the relation of inclusion.
Is has and B algebras, X generator parts of has , F and G of homomorphisms of has in B . If the restrictions of F and G on X are equal, then F and G is equal.
Is has and B algebras, F a homomorphism of has in B and X part of has . Then the image by F of the subalgebra of has generated by X is the subalgebra of B generated by F ( X ).
the subalgebra of an algebra has is the empty set is included in any subalgebra of has and, so that the empty set is a generating part of an algebra has , it is necessary and it is enough that there does not exist subalgebra of has different from has .

Products of algebras

The concept of product (direct) of algebras of groups, ring X and module S spreads within the general framework of the universal algebra.

That is to say $(A_i) _ {I \ in I}$ a family of algebras of signature Ω indexed by a unit (finished or not) and P the produced of the unit subjacent with these algebras.

Definition. There exists a single algebraic structure of signature Ω on P such as, for entire naturalness N and any element ω of $\ Omega_n$ , $\ omega_A$ ( $x_1$ ,…, $x_n$ ) = $(\ omega_ {A_i} (x_ {1i},…, x_ {nor}))_ {I \ in I}$ for any element $x_k$ = $(x_ {ki}) _ {I \ in I}$ of P , with K = 1,…, N . It is said that the algebra that is P provided with this algebraic structure is the direct produced or the produced or the algebra produced . of this family of algebras. It is noted $\ prod_ {I \ in I} A_i$ . If I = {1,…, N} , the note also $A_1$ ×… × $A_n$ .

Example. Prenons the case of two algebras has and B . Then, the structure of algebra of has × B is in the following way defined: for entire naturalness N , for any element ω of $\ Omega_n$ , whatever the elements $a_i$ of $A_i$ and $b_i$ of $B_i$ , with I = 1,…, N , one has $\ omega_ {has \ times B}$ (( $a_1$ , $b_1$ ),…, ( $a_n$ , $b_n$ )) = ( $\ omega_A$ ( $a_1$ ,…, $a_n$ ), ( $\ omega_B$ ( $b_1$ ,…, $b_n$ ).

Here some elementary properties of the products of algebras.

canonical projectors of $\ prod_ {I \ in I} A_i$ . (which with an element associates its component of index given) are surjective homomorphisms.
the algebraic structure of $\ prod_ {I \ in I} A_i$ east is the single algebraic structure on the unit $\ prod_ {I \ in I} A_i$ for which the canonical projectors are homomorphisms.
Is, for all I in I , $f_i$ a homomorphism of an algebra $A_i$ in an algebra $B_i$ . Then the application $\ prod_ {I \ in I} f_i$ of $\ prod_ {I \ in I} A_i$ in $\ prod_ {I \ in I} B_i$ which with $(x_i) _ {I \ in I}$ associates $(f_i (x_i))_ {I \ in I}$ is a homomorphism.
Is, for all I in I , $B_i$ a subalgebra of $B_i$ . Then $\ prod_ {I \ in I} B_i$ is a subalgebra of $\ prod_ {I \ in I} A_i$ .
the product of algebras is associative and commutative, in a direction to specify that the reader will be able to determine (in a way similar to the Cartesian product of units).
the graph of a homomorphism between two algebras has and B is a subalgebra of has × B .

Here the fundamental property of the products of algebras.

Theorem. Is $p_i$ (for all I in I ) the canonical projectors of the product $\ prod_ {I \ in I} A_i$ . and an algebra is B. Whatever homomorphisms $f_i$ of B in $A_i$ (for all I in I ), there exists a single homomorphism G of B in $\ prod_ {I \ in I} A_i$ such as, for all I in I , $p_i \ circ G = f_i$ (it is the homorphism whose components are the $f_i$ ).

The definite product corresponds to the product in terms of Théorie of the categories.

Algebra of the applications

Has an algebra and X are a unit. The whole of the applications of X in has is identified with the product $A^X$ algebras all equal to has . It is thus followed from there that there exists a canonical algebraic structure on the unit of the applications of X in has .

Here whatever properties of the algebras of the applications.

the diagonal application of has in $A^X$ (which with an element of has associates the corresponding constant application) is a homomorphism.

For any element X of X , the evaluation in X , which with an application of X in has associates its value X , is a homomorphism of $A^X$ in has (it is projector of the product).

For very part Y of X , the application of $A^X$ in $A^Y$ which with an application of X in has associates its restriction on Y is a homomorphism.

Are has and B algebras. The whole of homomorphisms of has in B is not necessarily a subalgebra of the algebra of the applications of has in B . However, it is the case for the commutative Monoïde S, the commutative groups, the Module S on a commutative Anneau, but not for the rings.

Quotients and congruences

Elementary definition and properties
It is current to define the Groupe quotient of a group by a Sous-groupe distinguished, the Anneau quotient of a ring by a Idéal bilatère or the module quotient of a module by a submodule. But the generalization of these concepts within the universal framework of the algebra is less immediate. These quotients are in fact of the quotients compared to particular relations of equivalence and in these examples, the classes of neutral equivalence of element ( E , 1 and 0 respectively) are the sub-groups distinguished, the ideals bilatères and the submodules.
Either has an algebra of signature Ω. One calls congruence (or Ω- congruence if one makes a point of specifying) has all Relation of equivalence R in has such as, for entire naturalness N , any element ω of $\ Omega_n$ whatever the elements $x_1$ ,…, $x_n$ , $y_1$ ,…, $y_n$ of then has, if, for I = 1,…, N , $x_i$ and $y_i$ are equivalent for R , $\ omega_A$ ( $x_1$ ,…, $x_n$ ) = $\ omega_A$ ( $y_1$ ,…, $y_n$ ) are equivalent for R .
Either R a congruence of has . Then each operation of has pass to the following quotient R , i.e. produced a “well defined” operation as a whole quotient has / R : the compound of the class of equivalence of elements of has for R is the class of equivalence of composed of these elements of has for R . One thus defines an algebraic structure on has / R , and the algebra thus obtained is called algebra quotient or, more simply, quotient of has by R . When one considers the unit has / R like an algebra of signature Ω, it is for this algebraic structure.
The canonical Surjection of has on has / R is a homomorphism. In fact, the algebraic structure of has / R is the single algebraic structure on the unit has / R for which this surjection is a homomorphism.
Let us reconsider the case of the groups. That is to say G a group. Then, for all Sub-group distinguished H from G , the relation of equivalence in G defined by H ( X and is equivalent there if and only if $xy^ {- 1}$ belongs to H ) is a congruence of G . Reciprocally, for any congruence R of G , the class of equivalence of the neutral element of G is a sub-group distinguished from G . One obtains thus a Bijection between congruences of G and the sub-groups distinguished from G . One obtains similar results for the rings and the modules by replacing the sub-groups distinguished by the ideals bilatères and the submodules, respectively.
Congruences of has are not other than the subalgebras of the algebra produced has × has .
Here some properties of congruences.

the identity of the unit has and the unit has × has are congruences of has .

the intersection of a whole of congruences of has is a congruence of has .

Is B a subalgebra of has . For any congruence R of has , the relation of equivalence in B induced by R (its intersection R ∩ ( B × B ) with B × B ) is a congruence in B , which is known as induced on B by R .

For any binary relation R in has , there exists a smaller congruence of has which contains it, and it is the intersection of the whole of congruences of has which contain it, and it is known as generated by R .

the whole of congruences of has is a complete lattice for the relation of inclusion, i.e. any whole of congruences of has admits, for the relation of inclusion, a limits lower and a limits higher. In this lattice, the lower limit of a family of congruences is her intersection and the limit upper is the congruence generated by its meeting.

Passage au quotient and theorems of isomorphism

As in the case of the groups, of the rings and the modules, there is a concept of passage to the quotient of homomorphisms and there are theorems of isomorphism.
First Theorem of isomorphism. Is has and B algebras, F a homomorphism of has in B . Then the relation of equivalence U in has associated with F ( X is equivalent to there if and only if F ( X ) = F ( there )) is a congruence in has , said associated with F and is noted $R_f$ , and the application G of has / $R_f$ in B deduced from F by passage to the quotient is an injective homomorphism, and thus a G defines an isomorphism of has / $R_f$ on F ( B ). If F is a Surjection, then G is an isomorphism of has / $R_f$ on B .
More generally, if R and S is congruences of has and B respectively and if, whatever the elements X and of has there which is equivalent for R , F ( X ) and F ( there ) is equivalent for S (it is said whereas F is compatible with R and S ), then, by passage to the quotient, one defines an application of has / R in B / S , and this application is a homomorphism.
Proposal. Is R a Relation of equivalence in an algebra has . So that R is a congruence of has , it is necessary and it is enough that there exists a homomorphism F of has in an algebra B such as R is the congruence of has associated with F .
Definition. One says that an algebra B is image homomorphic of an algebra has if there exists a surjective homomorphism F of has on B , and then B is isomorphous with the algebra has / R , where R is the congruence of has associated with F .
Here the fundamental property of the algebras quotients.
Theorem. Is has and B algebras, R a congruence of has and p the canonical surjection of has on has / R . For any homomorphism F of has / R in B , then F $\ circ$ p is a homomorphism of has in B which is compatible with R . The application F $\ mapsto$ F $\ circ$ p of the whole of homomorphisms of has / R in B in the whole of the homomorphism of has in B which are compatible with R is a bijection. In terms of Theory of the categories, that can be interpreted by saying that the couple ( has / B , p ) is Foncteur of the category of the algebras of signature Ω in the category of the units which with any algebra X of signature Ω associates the whole of homomorphisms of has in X which are compatible with R is representative.
Definition. Is has an algebra, B a subalgebra of has and R a congruence of has . It is said that B is saturated for R if the class of equivalence for R of very any element of B is included in B , i.e. if B is the meeting of classes of equivalence of R . The meeting of the whole of the classes of equivalence for R of the elements of B is a subalgebra of has which is saturated for R , and it is called saturated with B for R . It is equal to the image R ( B ) of B by the relation R .
Proposal. Is has and B algebras such as there exists a homomorphism F of has on B and is R the congruence of has associated with F . Then the application of the whole of the subalgebras of has which are saturated for R as a whole with the subalgebras with F ( B ) which with such a subalgebra C of has associates F ( C ) is a bijection.
Second theorem of isomorphism. Is has an algebra, B a subalgebra of has , R a congruence of has and C saturated with B for R . Then the canonical injection of B in C is compatible with the relation of equivalence S and T induced on B and C by R , and thus, one has by passage to the quotient, a homomorphism G of B / S in C / T . So moreover = C has, then G is an isomorphism of B / S on has / R = C / T .
Third theorem of isomorphism. Is has an algebra, R and S of congruences of has such as R is included in S . Then the identity of has is compatible with R and S and then the relation of equivalence associated with the application H with has / R on has / S deduced from the identity of has by passage to the quotient is noted S / R and is called quotient of S by R . The application of ( has / R )/( S / R ) in has / S deduced from H by passage to the quotient is an isomorphism (the R are simplified ).
Proposal. Is has an algebra and R a congruence of has . Then the application of the whole of congruences of has which contain R in the whole of congruences of has / R which with any congruence S of has which contains R associates S / R is a bijection.
Proposal. Is $(A_i) _ {I \ in I}$ a family of algebra and, for any element I of I , $R_i$ a congruence in $R_i$ . Then the binary Relation R in $\ prod_ {I \ in I} A_i$ defined by $(x_i) _ {I \ in I}$ R $(y_i) _ {I \ in I}$ if and only if $x_i$ R $y_i$ is a congruence in $\ prod_ {I \ in I} A_i$ . It is the relation of equivalence associated with homomorphism G with $\ prod_ {I \ in I} A_i$ in $\ prod_ {I \ in I} A_i/R_i$ which with $(x_i) _ {I \ in I}$ associates the $family (R_i (x_i))_ {I \ in I}$ of the classes of equivalence of the $x_i$ according to $R_i$ . According to the first theorem of isomorphism, one deduces from G by passage to the following quotient R an isomorphism of $\ prod_ {I \ in I} A_i /R$ on $\ prod_ {I \ in I} (A_i/R_i)$ .

Algebra of the terms

The algebras of the terms of signature Ω on a unit is an algebra which will make it possible to define the concept of identity in an algebra, for example associativeness, the commutatvity and the distributivity. Intuitively, it is made of all the formal combinations of elements of this whole starting from elements of Ω, interpreted like operators. One can think of this algebra like a kind of algebra of Polynôme S in unspecified (of finished or infinite number).
Definition. Is I a unit. There exists a smaller unit T such as I and $\ Omega_0$ is included in T (one supposes these two disjoined units) and such as, for entire naturalness not no one N , for any element ω of $\ Omega_n$ and whatever N elements $x_1$ ,…, $x_n$ of T , the continuation (ω, $x_1$ ,…, $x_n$ ) belongs to T . There then exists a single algebraic structure of signature Ω on T such as $\ Omega_0$ is the whole of the constants of T and such as, for entire naturalness not no one N , for any element ω of $\ Omega_n$ and whatever N elements $x_1$ ,…, $x_n$ of T , $\ omega_T$ ( $x_1$ ,…, $x_n$ ) = (ω, $x_1$ ,…, $x_n$ ), which makes it possible to indicate by ω ( $x_1$ ,…, $x_n$ ) the element (ω, $x_1$ ,…, $x_n$ ) of T . One calls Algèbre of the terms of signature Ω built on I and one notes $T_ {\ Omega}$ ( I ) or T ( I ) the algebra thus obtained. It is called also algebras of the words or absolutely free algebra . One calls terms or words the elements of T .
The element I of T is noted $X_i$ and the $X_i$ are called unspecified T .
Thus, the terms are formal expressions while making operated the elements of Ω on the unspecified ones ( $X_i$ , elements of I ) and the elements of the constants (elements of $\ Omega_0$ ), and while making operate the elements of preceded Ω on the expressions thus obtained and reiterating, a finished number of times.
Theorem . The algebra T = $T_ {\ Omega}$ ( I ) has a universal Propriété: for any algebra has and for any application F of the unit I in has, there exists a single homomorphism of algebras of T in has which prolongs F , and one obtains thus a Bijection between the whole of the applications of I in has and the whole of homomorphisms of T on in has .
Either has an algebra. By identifying the whole of the applications of I in has with the whole of the families of elements of has indexed by I , one has, according to what precedes, a canonical bijection φ between the whole of the families of elements of has indexed by I and the whole of homomorphisms of T in has . For any family $(a_i) _ {I \ in I}$ of elements of has indexed by I and for any element T of T , one notes note $t ((a_i) _ {I \ in I}$ the value in T of homomorphism φ ( $(a_i) _ {I \ in I}$ ) of T in has : it is said whereas this element of has is obtained in substitutant the $a_i$ with the $X_i$ . In an intuitive way, one replaces each occurrence of unspecified the $X_i$ in the term T by the element $a_i$ of has and one calculates the expression obtained in has . That is interpreted same manner as the substitition with unspecified of a Polynôme of element of a algebra unifère associative commutative on a commutative Anneau.
Let us note that, for any algebra has , it exists a surjective homomorphism with values in has defined on the algebra of the terms built on has .

Varieties of algebras

Identities

Definition. One calls identity of signature Ω built on X any couple of elements of T = $T_ {\ Omega}$ ( X ). Being given one identiré ( U , v ), it is said that an algebra has satisfied the identity U = v if, for any family X = $(x_i) _ {I \ in I}$ of elements of has indexed by I , one has U ( X ) = v ( X ), in other words, in substituent, for any element I of I , the same element of has in to unspecified the $X_i$ in U and v , the elements of has thus obtained are equal.
Examples. One considers a magma M , i.e. a unit provided with only one law of composition.

One supposes that I = {1, 2,3} and one will considéront T = T ({1, 2,3}) and the identity ( $x_1$ $x_2$ ) $x_3$ = $x_1$ ( $x_2$ $x_3$ ). To say that M satisfies this identity, it is to say that M is associative.

One supposes that I = {1, 2} and one will considéront T = T ({1, 2}) and the identity $x_1$ $x_2$ = $x_2$ $x_1$ . To say that M satisfies this identity, it is to say that M is commutative. This example also applies to the Monoïde S, the groups and the rings.

Examples. Here some examples of identities:

Associativeness of a Law of composition;

Commutation of a law of composition;

idempotence of a law of composition (the square of any element X is equal to X ), for example laws of a lattice;

Distributivité of a law of composition on another (case of a ring);

distributivity of a external Law of composition on a law of composition (case of the modules on a ring);

the exo-associativeness of an external law (case of the modules on a ring);

the fact that an element is a neutral element for a law of composition;

makes it that an element is exo-unifère for an external law (case of the modules (unifères) on a ring);

the fact that a given application associates with any element of a group its Inverse.

Varieties of algebras

Definition. a Variété of algebras of signature Ω is a class V of algebras of signature Ω such as it exists a unit I and a part S of $T_ \ Omega$ ( I ) × $T_ \ Omega$ ( I ) such as V is the class of all the algebras of signature Ω which satisfy each identity of S . In fact, one could, to in general define the varieties of algebras, to limit themselves to consider the algebra of the terms of a countable infinite unit fixed once for all (the whole of the natural entireties, for example).
For example, the monoids, the groups, the rings, the modules on a given ring (or vector spaces on a given body) form varieties of algebras.
The varieties of algebras of signature given form categories for homomorphisms and the composition of homomorphisms, and these categories have the majority of the usual common properties of the categories of the groups, the Monoïde S, the rings, the vector spaces on a body, etc: construction of the induced structures and the quotients, existence and construction of the products, existence of free objects, existence of the limits and the unspecified colimites, construction of the unspecified limits and the filter colimites. In a sense, one can say that the varieties of algebras are “good” categories of algebras.
The class of all the algebras which are empty or commonplace form a variety of algebras.
Certain operations of algebraic structures which make that one has to make with a variety of algebras given in the usual definition. For example, a group a unit provided with a law of composition checking certain properties, but only the existence of the neutral element and the existence of the Inverse of any element belong to the usual definition, but the group, in the usual definition, is not provided with a neutral element and an inversion. To determine the operations which whose that one has to make with a variety of algebras, one can examine the axioms of substructures (sub-groups, subrings, etc).
That is to say V a variety of algebras. There is a Foncteur, known as of lapse of memory , category V in the category of the whole while associating with any algebra has V his subjacent unit.

Examples of varieties of algebras

For each one of the type of algebraic structure following, the class of all the algebras which are of this type form a variety of algebras:

units (with any operation finitaire);

the pointed units, i.e. units provided with an element;

the magmas, i.e. units provided with a law of composition;

associative magmas, i.e. the magmas whose law is associative;

commutative magmas, i.e. the magmas whose law is commutative;

unit magmas, i.e. magmas having an element unit;

the Monoid S, i.e. the unit magmas whose law is associative;

commutative monoids;

the groups;

commutative groups

the G - units, where G is a group (or a monoid), i.e. the units E provided with a action of G on E ;

the Quasigroupe S;

the Semi-ring X;

the rings;

the commutative rings;

the modules on a given ring has (in particular, the vector spaces on a body given K);

the algebras on a given commutative ring has (without assumptions on the multiplication other than the bilinearity);

the associative algebras on a given commutative ring has;

the associative unit algebras on a given commutative ring has;

the commutative associative unit algebras a given commutative ring has;

the algebras of Dregs on a given commutative ring has;

the algebras of Jordan on a commutative ring given has ;

the lattice;

distributive lattices;

modular lattices;

the Boolean algebra.

Here algebraic structures which do not form varieties of algebras: the Semigroup S, i.e. the monoids of which any element is that can be simplified, the body, the principal rings, the factorial rings, the free modules on a given ring has which is not a body.
Either X a Espace refines on a body K , one regarding the empty set as a space refines attached to a vector space no one. Then, for entire naturalness not no one N and for any finished continuation of N elements of K whose sum is equal to 1, one has operation N - surface which with a continuation of N - points associates the Barycentre these points affected of these elements of K . One thus defines an algebraic structure on X . If one gives oneself another space refines Y on K , the applications closely connected of X in Y are not other than homomorphisms for these algebraic structures. Moreover, the subspaces closely connected of X (including the empty set) are not other than the subalgebra for this algebraic structure. In fact, one can show, that, while associating with each space refines on K the algebra thus defined, one has a functor of the category of space closely connected on K (the morphisms are the applications closely connected) in a variety of algebras which is in fact a equivalence of categories. This shows that the properties catégorielles of the varieties of algebras apply to the category of spaces closely connected to K .

Properties of the varieties of algebras

The varieties of algebras are stable for the majority of usual constructions in algebra. That is to say V a variety of algebras. There are the following properties.

Any subalgebra of an algebra of V is an algebra of V .

For any congruence R of an algebra has V , the algebra quotient of has by R is an algebra of V .

Whatever the algebras has and B , if there exists a surjective homomorphism of has on B and if has is an algebra of V , then B is an algebra of V .

Any algebra which is isomorphous with an algebra of V is an algebra of V .

the algebra produces of a family of algebras of V is an algebra of V .

the empty algebra (if there are some) is an algebra of V and any commonplace algebra (i.e. which are a singleton) is an algebra of V .

In fact, one with the following characterization of the varieties of algebras.
Theorem of Birkhoff. So that a class V of algebras of signature Ω is a variety of algebras, it is necessary and it is enough that it checks the following properties:

any subalgebra of an algebra of V is an algebra of V ;

whatever the algebras has and B , if has is an algebra of V and if there exists a surjective homomorphism of has on B , then B is an algebra of V ;

the product of a family of algebras of V is an algebra of V .

Proposal. Is V a variety of algebras. Then, within the meaning of the Theory of the categories, the Isomorphisme S of the category V are not other than homomorphisms of V which are bijections and the Monomorphisme S of V are not other than homomorphisms of V which are injections. Any surjective homomorphism of V is a epimorphism of the category V , but the reciprocal one can be false, as the category of the rings shows it (or of the monoids): one can show that there exists, in the category of the rings, a nonsurjective epimorphism of the ring Z of the rational entireties in the body Q of the rational numbers.
In the varieties of algebras, certain authors call epimorphisms surjective homomorphisms, which can created a confusion with the epimorphism of the theory of the categories.
Proposal. the direct products of algebras of V are not other than the products within the meaning of the Théorie of the categories.
ALL the general properties of the varieties of algebras apply to all the algebraic structures which form varieties of algebras enumerated previously, and to good of others. One can thus define for his structures of homomorphisms, the subalgebras and congruences, and one can built the direct products and the quotients, and this, in a uniform way. Moreover, as it will be seen, one can build the unspecified limits and the filter colimites of functors (in particular of the projective limiting and inductive of projective systems and systems inductive indexed by filter ordered units) in the same way as in Set theory. In particular there are the equalizing ones, coégalisateurs and products fibers, built as in set theory. One can as show as these structures admit unspecified colimites, and in particular coproduits (or sums) and coproduits fibers (or sums ammalgamées or fibrées), but each variety of algebras to its construction, which can differ from construction that one finds in set theory. One can build the free algebras generated by units.

Free algebras

Free algebras
One finds in different algebras free objects on a unit: free monoids, the free groups, the free modules on a given ring, algebras of Polynomial S built with coefficients in a commutative ring. All these examples are rather well-known (at the university level). All these constructions have common properties, which in fact concerns the Théorie of the categories, and spread in the varieties of algebras.
That is to say V a variety of algebra of signature Ω, and it is supposed that there exist noncommonplace algebras in V , i.e. who have more than one elements. Then there is the following theorem:
Theorem. For any unit I , there exists an algebra L of V which contains I like subset such as, for any algebra has V and for any application F of I in has , it exists a single homomorphism of L in has which prolongs F . It is said that such an algebra L is a free algebra built on I in V . The part I of L is then a generating part of L .
If V is the variety of all the algebras of signature Ω, then the algebra of terms $T_ {\ Omega}$ ( I ) is a free algebra built on I .
Proposal. Whatever the free algebras L and M built on I in V , there exists a single isomorphism of L on M which prolongs the identity application of the unit I . Thus, the free algebra built on I in V is single except for a single isomorphism.
The free algebras built on I in V are thus single except for a single isomorphism. One can thus choose some once for all. But there is of it a canonical construction, which is an algebra quotient of the algebra of the terms built on I . Here its construction. For any free algebra built L on I in V , there exists a single homomorphism of the algebra of the terms T = $T_ {\ Omega}$ ( I ) in L which prolongs the canonical injection of I in L and it is surjective and moreover the relation of equivalence R in T associated with this homomorphism depends only on V and I . One thus obtains an isomorphism of T / R on L . Moreover the made up one of the canonical injection of I in T and of the canonical surjection of T on T / I is injective, and thus, by identifying then I with its image in T / I for this injection, T / I is a free algebra built on I in V . It is called the free algebra built on I in V and one notes it $L_V$ ( I ).
Proposal. Whatever the units I and J and the application F of I in J , there exists a single homomorphism of $L_V$ ( I ) in $L_V$ ( J ) which prolongs F , and it is noted $L_V$ ( F ). This homomorphism is injective, surjective or an isomorphism according to whether F is injective, surjective and bijective. In terms of the Theory of the categories, one definite thus a functor F of the category of the whole in V , which is in fact an assistant on the left of the functor of lapse of memory of V in the category of the units.
Definition. One says that an algebra of V is free (without reference to a whole of unspecified) if it is isomorphous a free algebra built on a whole in V . That generalizes the free concept of Module builds on a unit.
For any algebra has , it exists a surjective homomorphism with values in has defined on a free algebra built on the unit subjacent with has .

Substitution

That is to say I a unit. For any element I of I , one can then note $X_i$ the element of $L_V$ ( I ) to which identife I , and one says that the $X_i$ are unspecified of $L_V$ ( I ). By thus intrerprétant the elements of I , there is the result which follows.
Either has an algebra of V . By identifying the whole of the applications of I in has with the whole of the families of elements of has indexed by I , one has, according to the definition of the free algebras, a canonical bijection φ between the whole of the families of elements of has indexed by I and the whole of homomorphisms of L = $L_V$ ( I ) in has . For any family $(x_i) _ {I \ in I}$ of elements of has indexed by I and for any element T of L , one notes $t ((x_i) _ {I \ in I})$ the value in T of φ ( $(x_i) _ {I \ in I}$ ): it is said whereas this element of has is obtained in substitutant the $x_i$ with the $X_i$ . That is interpreted same manner as the substitition with unspecified of a Polynôme of element of a commutative algebra unit associative on a commutative Anneau.
Either has an algebra of V , T an element of L = $L_V$ ( I ). By associating this manner with any element of $A^I$ an element of element of has , obtained by substitution in T , one obtains an application of $A^I$ in has which is interpreted, in the case of the variety of the commutative associative unit algebras on a commutative ring K , like the polynomial application associated with T .

Dependence and bases

Are has an algebra of V , $(x_i) _ {I \ in I}$ a family of elements of has indexed by a unit I and φ the homomorphism corresponding of $L_V$ ( I ) in has .

So that this family is a generating family of has , it is necessary and it is enough that φ is a Surjection, which depends only on the signature of algebras, and not of the variety of algebras.

One says that this family is free or that its elements are independent in has if φ is a injection, and if not one says that the family is dependant or that its dependant elements . That generalizes the linear concept of Dépendance which one meets in Linear algebra and of algebraic Dépendance that one meets in the commutative associative unit algebras on a commutative Anneau (algebra of the Polynôme S).

One says that a family of elements of has is a bases has if it is a generating family and a free family, i.e. if φ is an isomorphism. That generalizes the bases of the modules or the vector spaces.

the algebra has is free in V if and only if she admits a base relative with V .

Concepts of free family, elements independent and basic dépenend of the variety of algebras and not only of the signature of algebras, contrary with the case of the generating families.

Limits and colimites
Either V a variety of algebras of signature Ω. One can show that the category V admits limiting projective and limiting inductive, and even, more generally, limiting and colimites. One can as show as the functor of lapse of memory of V in the category of the units creates the unspecified limits and creates the filter colimites (filter inductive limits).
Theorem. Is I small a category (resp. a Together ordered filter) and F a Functor of I in V and will considéront L the limit (resp. the colimite) of F regarded as functor in the category of the units, by means of the functor of lapse of memory. Then there exists a single algebraic structure of signature Ω on L for which, for any element I of I , the canonical application of L in $F_i$ (resp. of L in $F_i$ ) is a homomorphism. Then, for this algebraic structure on L , L is, for the canonical applications, a limit (resp. a colimite) of the functor F in the category V . One can show that this construction is fonctorielle.
In particular, one can replace I by a Ensemble ordered (resp. a filter ordered unit) and F by a projective System (resp. a inductive System) of algebras of V : one obtains the projective limits (resp. inductive limits).
Examples (all built as in the category of the units)

the product of algebras of V .

the equalizing one and the coegalisator and of two homomorphisms between two algebras has and B .

the products fibers of algebras of has and B of V relative to homomorphisms U and v of an algebra S with values in has and B respectively.

The category V admits unspecified limits and colimites, and thus the category is complete and cocomplète. That would fall at fault if the algebras were not admitted (if there were some at least).
That has as a consequence which the varieties of algebras admit of the Coproduit S (or direct sums). Here example of coproduits:

For the families of groups (or the monoids), the coproduits are the free produced ;

For the families of modules on the same ring (or the groups commutative), the coproduits are the direct sums;

For the families finished of commutative associative unit algebras on same a commutative Ring, the coproduits is the tensorial products (in the same way for the commutative rings).

As it is seen, the coproduits are seldom built as in the case of the units (it is then the disjoined meeting).
The varieties of algebras admit also coproduits fibers (or ammalgamées sums).
Within the meaning of the theory of the categories, any variety of algebras has admits initial objects, i.e. an algebra U such as, for any algebra of has V , it exists a single homomorphism of U in has . The initial objects are, in the same variety of algebras, single except for a single isomorphism. In the category of the units, the empty set is the single initial object, but in a variety of algebras, the initial objects do not have like subjacent unit the empty set, except if there is no operation nullaire in the signature (whereas the final objects have as together subjacent singletons them, final objects of the category of the units). Here whatever examples of initial objects in varieties of algebras:

for the associative magmas or magmas, the empty magma is the single initial object;

for the units pointed, the monoids, the groups and the modules on a given ring, the initial objects are the commonplace algebras;

for the rings, the ring Z of the rational entireties in is an initial object;

For any commutative ring K , K is an initial object of the variety of the associative algebras unifères on K ;

the empty lattice is the single initial object of the variety of the lattices;

the Boolean algebra {0, 1} is an initial object of the variety of the Boolean algebra.

History

See too

Algèbre

general Algèbre

Law of composition

Law of composition interns

external Law of composition

algebraic Structure

References

Bergman, George Mr.: Year Invitation to General Algebra and Universal Constructions , Henry Helson, 1998. ISBN 0-9655211-4-1.

Burris, Stanley NR., and H.P. Sankappanavar: has Race in Universal Algebra Springer-Verlag, 1981. ISBN 3-540-90578-2.

Cohn, Paul Moritz, 1981. Universal Algebra . Dordrecht, Netherlands: D.Reidel Publishing. ISBN 90-277-1213-1.

Grillet, Pierre: Algebra , Wiley-Interscience Publication, 1999. ISBN 0-471-25243-3.

Jacobson, Nathan: BASIC Algebra II , W.H. Freeman, 1989. ISBN 0-7167-1933-9.

Random links: Georges Bess | Shirakawa (Japan) | Local union of company | Guillaume of Prussia (1783-1851) | Park of Docelles-Cheniménil