|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], one can often define a '''direct product''' of objects
| | There are a variety of ways to get rid of fat plus move forward with healthy living. Getting there is quite difficult, if youre not employed for you to get by the different items which are needed to move forward. If youre looking to lose weight, get fit plus possibly even build lean muscle, youll have to follow 3 major tricks. One of them includes Absonutrix Raspberry Ketones, that is functioning like a miracle. Consider the following 3 tricks which you can commence utilizing now to receive the maximum workout plus wellness program possible.<br><br>No side impact of the miracle fat burner has been reported thus far. However, create sure that you purchase a genuine supplement which contains all 8 elements stated above.<br><br>raspberry ketone supplements are not for everyone. It is not recommended to buy plus start utilizing any such supplement without consulting a doctor. Just search for "raspberry ketone reviews" plus we will be surprised to see which how badly these supplements have affected people's health. Although not all of the time, yet inside almost all of the cases they do. So the initially thing you require to do, whenever you think of using any such supplement, is to consult a doctor plus see what he/she has to suggest we, and do how you're suggested.<br><br>There is evidence which it could make you feel fuller faster. This is as a result of its fiber content. It may assist prevent cravings due to the spike of blood sugars. Finally, it can speed up one's body ability to burn fat.<br><br>[http://safedietplansforwomen.com/raspberry-ketones raspberry ketones] Plus recently hit the market inside the form of weight reduction medications. It became an instant hit following Dr Oz suggested it because a miracle fat-burner in a bottle. Raspberry Ketones Plus contains 8 all-natural elements that work wonders for burning fat naturally.<br><br>+ Weight LossThis product helps inside weight loss by controlling the metabolism of the body. It contains the adiponectin hormone (insulin sensitizing hormone) which regulates the metabolism of lipids and glucose. To understand it in raspberry ketone diet a easy code - the high amount of adiponectin hormone ensures the faster burning of the fats. This product concentrates on breaking down the unwelcome fat of the body, providing we the quicker fat reduction results.<br><br>Not numerous fat burners contain so many natural elements. Though there are fat reduction diet medications that contain one or 2 of above revealed components, they frequently function on one aspect of your health. A perfect blend of all of these elements is important for fast weight reduction without any negative effects.<br><br>Every of these eight factors might become the cause for a obese. Or even more likely, it's a combination of several of them. And it's crucial for the success to locate out the exact reasons. A plain diet won't assist you if the contraception system plus deficiency of activity are furthermore contributing a remarkable deal to the issue. To resolve a problem, we first have to define its cause. When you learn what caused your obese, you may be found on the best technique to lucrative permanent weight loss. |
| already known, giving a new one. This is generalize the [[Cartesian product]] of the underlying sets, together with a suitably defined structure on the product set.
| |
| More abstractly, one talks about the [[Product (category theory)|product in category theory]], which formalizes these notions.
| |
| | |
| Examples are the product of sets (see [[Cartesian product]]), groups (described below), the [[product of rings]] and of other [[abstract algebra|algebraic structures]]. The [[product topology|product of topological spaces]] is another instance.
| |
| | |
| There is also the [[direct sum]] – in some areas this is used interchangeably, in others it is a different concept.
| |
| | |
| == Examples ==
| |
| | |
| * If we think of <math>\mathbb{R}</math> as the [[set (mathematics)|set]] of real numbers, then the direct product <math>\mathbb{R}\times \mathbb{R}</math> is precisely just the [[cartesian product]], <math>\{ (x,y) | x,y \in \mathbb{R} \}</math>.
| |
| | |
| * If we think of <math>\mathbb{R}</math> as the [[group (mathematics)|group]] of real numbers under addition, then the direct product <math>\mathbb{R}\times \mathbb{R}</math> still consists of <math>\{ (x,y) | x,y \in \mathbb{R} \}</math>. The difference between this and the preceding example is that <math>\mathbb{R}\times \mathbb{R}</math> is now a group. We have to also say how to add their elements. This is done by letting <math>(a,b) + (c,d) = (a+c, b+d)</math>.
| |
| | |
| * If we think of <math>\mathbb{R}</math> as the [[ring (mathematics)|ring]] of real numbers, then the direct product <math>\mathbb{R}\times \mathbb{R}</math> again consists of <math>\{ (x,y) | x,y \in \mathbb{R} \}</math>. To make this a ring, we say how their elements are added, <math>(a,b) + (c,d) = (a+c, b+d)</math>, and how they are multiplied <math>(a,b) (c,d) = (ac, bd)</math>.
| |
| | |
| * However, if we think of <math>\mathbb{R}</math> as the [[field (mathematics)|field]] of real numbers, then the direct product <math>\mathbb{R}\times \mathbb{R}</math> does not exist – naively defining <math>\{ (x,y) | x,y \in \mathbb{R} \}</math> in a similar manner to the above examples would not result in a field since the element <math>(1,0)</math> does not have a multiplicative inverse.
| |
| | |
| In a similar manner, we can talk about the product of more than two objects, e.g. <math>\mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}</math>. We can even talk about product of infinitely many objects, e.g. <math>\mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \dotsb</math>.
| |
| | |
| == Group direct product ==
| |
| {{main|Direct product of groups}}
| |
| In [[group (mathematics)|group theory]] one can define the direct product of two
| |
| groups (''G'', *) and (''H'', ●), denoted by ''G'' × ''H''. For [[abelian group]]s which are written additively, it may also be called the [[Direct sum of groups|direct sum of two groups]], denoted by <math>G \oplus H</math>.
| |
| | |
| It is defined as follows:
| |
| * the [[Set (mathematics)|set]] of the elements of the new group is the ''[[cartesian product]]'' of the sets of elements of ''G'' and ''H'', that is {(''g'', ''h''): ''g'' in ''G'', ''h'' in ''H''};
| |
| * on these elements put an operation, defined elementwise: <center>(''g'', ''h'') × (''g' '', ''h' '') = (''g'' * ''g' '', ''h'' ● ''h' '')</center>
| |
| (Note the operation * may be the same as ●.)
| |
| | |
| This construction gives a new group. It has a [[normal subgroup]]
| |
| [[isomorphic]] to ''G'' (given by the elements of the form (''g'', 1)),
| |
| and one isomorphic to ''H'' (comprising the elements (1, ''h'')).
| |
| | |
| The reverse also holds, there is the following recognition theorem: If a group ''K'' contains two normal subgroups ''G'' and ''H'', such that ''K''= ''GH'' and the intersection of ''G'' and ''H'' contains only the identity, then ''K'' is isomorphic to ''G'' x ''H''. A relaxation of these conditions, requiring only one subgroup to be normal, gives the [[semidirect product]].
| |
| | |
| As an example, take as ''G'' and ''H'' two copies of the unique (up to
| |
| isomorphisms) group of order 2, ''C''<sub>2</sub>: say {1, ''a''} and {1, ''b''}. Then ''C''<sub>2</sub>×''C''<sub>2</sub> = {(1,1), (1,''b''), (''a'',1), (''a'',''b'')}, with the operation element by element. For instance, (1,''b'')*(''a'',1) = (1*''a'', ''b''*1) = (''a'',''b''), and (1,''b'')*(1,''b'') = (1,''b''<sup>2</sup>) = (1,1).
| |
| | |
| With a direct product, we get some natural [[group homomorphism]]s for free: the projection maps
| |
| :<math>\pi_1: G \times H \to G\quad \text{by} \quad \pi_1(g, h) = g</math>,
| |
| :<math>\pi_2: G \times H \to H\quad \text{by} \quad \pi_2(g, h) = h</math>
| |
| called the '''coordinate functions'''.
| |
| | |
| Also, every homomorphism ''f'' to the direct product is totally determined by its component functions
| |
| <math>f_i = \pi_i \circ f</math>.
| |
| | |
| For any group (''G'', *), and any integer ''n'' ≥ 0, multiple application of the direct product gives the group of all ''n''-[[tuple]]s ''G''<sup>''n''</sup> (for ''n'' = 0 the trivial group). Examples:
| |
| *'''Z'''<sup>''n''</sup>
| |
| *'''R'''<sup>''n''</sup> (with additional [[vector space]] structure this is called [[Euclidean space]], see below)
| |
| | |
| == Direct product of modules ==
| |
| The direct product for [[module (mathematics)|modules]] (not to be confused with the [[Tensor product of modules|tensor product]]) is very similar to the one defined for groups above, using the [[cartesian product]] with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from '''R''' we get [[Euclidean space]] '''R'''<sup>''n''</sup>, the prototypical example of a real ''n''-dimensional vector space. The direct product of '''R'''<sup>''m''</sup> and '''R'''<sup>''n''</sup> is '''R'''<sup>''m'' + ''n''</sup>.
| |
| | |
| Note that a direct product for a finite index <math>\prod_{i=1}^n X_i </math> is identical to the [[Direct sum of modules|direct sum]] <math>\bigoplus_{i=1}^n X_i </math>. The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of [[category theory]]: the direct sum is the [[coproduct]], while the direct product is the product.
| |
| | |
| For example, consider <math>X=\prod_{i=1}^\infty \mathbb{R} </math> and <math>Y=\bigoplus_{i=1}^\infty \mathbb{R}</math>, the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in ''Y''. For example, (1,0,0,0,...) is in ''Y'' but (1,1,1,1,...) is not. Both of these sequences are in the direct product ''X''; in fact, ''Y'' is a proper subset of ''X'' (that is, ''Y'' ⊂ ''X'').
| |
| | |
| == Topological space direct product ==
| |
| The direct product for a collection of [[topological space]]s ''X<sub>i</sub>'' for ''i'' in ''I'', some index set, once again makes use of the Cartesian product
| |
| | |
| :<math>\prod_{i \in I} X_i. </math>
| |
| | |
| Defining the [[topology]] is a little tricky. For finitely many factors, this is the obvious and natural thing to do: simply take as a [[basis (topology)|basis]] of open sets to be the collection of all cartesian products of open subsets from each factor:
| |
| | |
| :<math>\mathcal B = \{ U_1 \times \cdots \times U_n\ |\ U_i\ \mathrm{open\ in}\ X_i \}.</math>
| |
| | |
| This topology is called the [[product topology]]. For example, directly defining the product topology on '''R'''<sup>2</sup> by the open sets of '''R''' (disjoint unions of open intervals), the basis for this topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual [[metric space|metric]] topology).
| |
| | |
| The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous (i.e. to satisfy the categorical definition of product: the morphisms here are continuous functions): we take as a basis of open sets to be the collection of all cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many of the open subsets are the entire factor:
| |
| | |
| :<math>\mathcal B = \left\{ \prod_{i \in I} U_i\ \Big|\ (\exists j_1,\ldots,j_n)(U_{j_i}\ \mathrm{open\ in}\ X_{j_i})\ \mathrm{and}\ (\forall i \neq j_1,\ldots,j_n)(U_i = X_i) \right\}.</math>
| |
| | |
| The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, and this does yield a somewhat interesting topology, the [[box topology]]. However it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous (see the separate entry box topology for an example and more). The problem which makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is only guaranteed to be open for finitely many sets in the definition of topology.
| |
| | |
| Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called [[Tychonoff's theorem]], is yet another equivalence to the [[axiom of choice]].
| |
| | |
| For more properties and equivalent formulations, see the separate entry [[product topology]].
| |
| | |
| == Direct product of binary relations ==
| |
| On the Cartesian product of two sets with [[binary relation]]s ''R'' and ''S'', define (''a'', ''b'') T (''c'', ''d'') as ''a'' ''R'' ''c'' and ''b'' ''S'' ''d''. If ''R'' and ''S'' are both [[reflexive relation|reflexive]], [[irreflexive relation|irreflexive]], [[transitive relation|transitive]], [[symmetric relation|symmetric]], or [[antisymmetric relation|antisymmetric]], relation ''T'' has the same property.<ref>[http://cr.yp.to/2005-261/bender1/EO.pdf Equivalence and Order]</ref> Combining properties it follows that this also applies for being a [[preorder]] and being an [[equivalence relation]]. However, if ''R'' and ''S'' are [[total relation]]s, ''T'' is in general not.
| |
| | |
| == Categorical product ==
| |
| {{Main|Product (category theory)}}
| |
| | |
| The direct product can be abstracted to an arbitrary [[category theory|category]]. In a general category, given a collection of objects ''A<sub>i</sub>'' ''and'' a collection of [[morphism]]s ''p<sub>i</sub>'' from ''A'' to ''A<sub>i</sub>''{{clarify|Is A a single object from A_i, or all A_i?|date=February 2012}} with ''i'' ranging in some index set ''I'', an object ''A'' is said to be a '''categorical product''' in the category if, for any object ''B'' and any collection of morphisms ''f<sub>i</sub>'' from ''B'' to ''A<sub>i</sub>'', there exists a unique morphism ''f'' from ''B'' to ''A'' such that ''f<sub>i</sub> = p<sub>i</sub> f'' and this object ''A'' is unique. This not only works for two factors, but arbitrarily (even infinitely) many.
| |
| | |
| For groups we similarly define the direct product of a more general, arbitrary collection of groups ''G<sub>i</sub>'' for ''i'' in ''I'', ''I'' an index set. Denoting the cartesian product of the groups by ''G'' we define multiplication on ''G'' with the operation of componentwise multiplication; and corresponding to the ''p<sub>i</sub>'' in the definition above are the projection maps
| |
| | |
| :<math>\pi_i \colon G \to G_i\quad \mathrm{by} \quad \pi_i(g) = g_i</math>,
| |
| | |
| the functions that take <math>(g_j)_{j \in I}</math> to its ''i''th component ''g<sub>i</sub>''. | |
| <!-- this is easier to visualize as a [[commutative diagram]]; eventually somebody should insert a relevant diagram for the categorical product here! -->
| |
| | |
| == Internal and external direct product ==
| |
| <!-- linked from [[Internal direct product]] and [[External direct product]] -->
| |
| {{see also|Internal direct sum}}
| |
| | |
| Some authors draw a distinction between an '''internal direct product''' and an '''external direct product.''' If <math>A, B \subset X</math> and <math>A \times B \cong X</math>, then we say that ''X'' is an ''internal'' direct product (of ''A'' and ''B''); if ''A'' and ''B'' are not subobjects, then we say that this is an ''external'' direct product.
| |
| | |
| ==Metric and norm==
| |
| A metric on a Cartesian product of metric spaces, and a norm on a direct product of normed vector spaces, can be defined in various ways, see for example [[Norm_%28mathematics%29#p-norm|p-norm]].
| |
| | |
| ==See also==
| |
| *[[Direct sum]]
| |
| *[[Cartesian product]]
| |
| *[[Coproduct]]
| |
| *[[Free product]]
| |
| *[[Semidirect product]]
| |
| *[[Zappa–Szep product]]
| |
| *[[Tensor product of graphs]]
| |
| *[[Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets|Orders on the Cartesian product of totally ordered sets]]
| |
| | |
| == Notes ==
| |
| <references />
| |
| | |
| == References ==
| |
| *{{Lang Algebra}}
| |
| | |
| {{DEFAULTSORT:Direct Product}}
| |
| [[Category:Abstract algebra]]
| |
| | |
| [[ru:Прямое произведение#Прямое произведение групп]]
| |
There are a variety of ways to get rid of fat plus move forward with healthy living. Getting there is quite difficult, if youre not employed for you to get by the different items which are needed to move forward. If youre looking to lose weight, get fit plus possibly even build lean muscle, youll have to follow 3 major tricks. One of them includes Absonutrix Raspberry Ketones, that is functioning like a miracle. Consider the following 3 tricks which you can commence utilizing now to receive the maximum workout plus wellness program possible.
No side impact of the miracle fat burner has been reported thus far. However, create sure that you purchase a genuine supplement which contains all 8 elements stated above.
raspberry ketone supplements are not for everyone. It is not recommended to buy plus start utilizing any such supplement without consulting a doctor. Just search for "raspberry ketone reviews" plus we will be surprised to see which how badly these supplements have affected people's health. Although not all of the time, yet inside almost all of the cases they do. So the initially thing you require to do, whenever you think of using any such supplement, is to consult a doctor plus see what he/she has to suggest we, and do how you're suggested.
There is evidence which it could make you feel fuller faster. This is as a result of its fiber content. It may assist prevent cravings due to the spike of blood sugars. Finally, it can speed up one's body ability to burn fat.
raspberry ketones Plus recently hit the market inside the form of weight reduction medications. It became an instant hit following Dr Oz suggested it because a miracle fat-burner in a bottle. Raspberry Ketones Plus contains 8 all-natural elements that work wonders for burning fat naturally.
+ Weight LossThis product helps inside weight loss by controlling the metabolism of the body. It contains the adiponectin hormone (insulin sensitizing hormone) which regulates the metabolism of lipids and glucose. To understand it in raspberry ketone diet a easy code - the high amount of adiponectin hormone ensures the faster burning of the fats. This product concentrates on breaking down the unwelcome fat of the body, providing we the quicker fat reduction results.
Not numerous fat burners contain so many natural elements. Though there are fat reduction diet medications that contain one or 2 of above revealed components, they frequently function on one aspect of your health. A perfect blend of all of these elements is important for fast weight reduction without any negative effects.
Every of these eight factors might become the cause for a obese. Or even more likely, it's a combination of several of them. And it's crucial for the success to locate out the exact reasons. A plain diet won't assist you if the contraception system plus deficiency of activity are furthermore contributing a remarkable deal to the issue. To resolve a problem, we first have to define its cause. When you learn what caused your obese, you may be found on the best technique to lucrative permanent weight loss.