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| Edward Green |
 Posted: Sat Dec 20, 2003 9:01 am |
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The concept of "generator" is popular in physics and mathematics.
Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity... |
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| Martin Bundgaard |
| Posted: Sat Dec 20, 2003 9:27 am |
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Edward Green wrote:
[quote:002f129e81]Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity...
[/quote:002f129e81]
That is because many interesting objects in physics are "groups".
Generators are distinguished elements of the group from which all other
elements can be constructed.
So what you are looking for is "group theory". Google has a vast amount
of material to offer.
Martin
--
"Tout ce qu'il y a de bébête." -- Grothendieck |
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| Tobias Fritz |
| Posted: Sat Dec 20, 2003 9:52 am |
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[quote:e8fb8d54cf]
That is because many interesting objects in physics are "groups".
Generators are distinguished elements of the group from which all other
elements can be constructed.
So what you are looking for is "group theory". Google has a vast amount
of material to offer.
but the word "generator" is not only used for groups, but for example also[/quote:e8fb8d54cf]
for modules where some elements may generate the module as a module, but
not as an abelian group.
Is the word also commonly used in topology for a subbase?
--
Physics is much too hard for physicists.
reverse my forename for mail! |
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| Martin Bundgaard |
| Posted: Sat Dec 20, 2003 11:51 am |
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Tobias Fritz wrote:
[quote:b57b2cd1ad]Is the word also commonly used in topology for a subbase?
[/quote:b57b2cd1ad]
Well, you do have subsets of a topology which generate it (by unions and
intersections), but the word "generator" is afaik not used in this
context. Rather one says that the set generates the topology or that it
is a "pre-topology".
Martin
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"Tout ce qu'il y a de bébête." -- Grothendieck |
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| Leonard Blackburn |
| Posted: Sat Dec 20, 2003 4:56 pm |
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nulldev00@aol.com (Edward Green) wrote in message news:<2a0cceff.0312200601.4d6cadbd@posting.google.com>...
[quote:0f8de50848]The concept of "generator" is popular in physics and mathematics.
Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity...
[/quote:0f8de50848]
I believe the unifying concept is the theory of inductive definability. |
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| Lee Rudolph |
| Posted: Sat Dec 20, 2003 5:43 pm |
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blackbur@math.umn.edu (Leonard Blackburn) writes:
[quote:9eb163da8f]nulldev00@aol.com (Edward Green) wrote in message news:<2a0cceff.0312200601.4d6cadbd@posting.google.com>...
The concept of "generator" is popular in physics and mathematics.
Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity...
I believe the unifying concept is the theory of inductive definability.
[/quote:9eb163da8f]
I believe you believe wrongly (but if you're coming from sci.physics,
who knows?).
First of all (this is not responsive to LB, but rather to another
respondent to EG, and to EG himself), it's rare (though not unheard
of) for a mathematical structure (of interest) with "generators" to
have _a_ generator (or a set of generators) which is particularly
unique; and often it's as bad mathematically to be fixated on
particular generators as it is to be fixated on a particular
coordinate system. (Indeed, "generators" can often be thought
of as nearly the same thing as a "coordinate system": for instance,
in a vectorspace, an irredundant set of generators is precisely the
same thing as an unordered basis, while a coordinate system is
essentially the same thing as an ordered basis.)
As it's most often used (in the fields I'm familiar with),
a subset X of a structure Y of a given type (say, a group,
or a vectorspace, or an algebra; but also less algebraic
things) would be said to "generate Y" if any substructure
of Y of the given type, which contains X, is all of Y.
*Sometimes* there will then be a way to "present" Y using the
elements of X as "generators" subject to certain "relations"
(Arturo Magidin is the local expert in universal algebra, and
may know a much more general context for this than what I'm
about to say): one can (in favorable cases) consider the "free"
structure F of the same type of Y "freely generated" by X,
represent Y as a "quotient structure" F/R, and interpret each
element of R as a "relation" among the elements of X. Even
then, however, I think it's an error to believe that there's
some sort of "inductive definability" going on, necessarily.
(There are all sorts of fine logical points around what "definability",
or "inductive" for that matter, might mean here, so maybe it's
not an error.)
Minimal sets of generators (when they exist) are likely to be
more interesting than random sets of generators (in the same style
that a basis of a vectorspace is more interesting than simply
a spanning set in the vectorspace). "Non-generators"--elements of
Y which belong to *no* minimal set of generators--can be interesting,
too; maybe even more than "generators" (non-non-generators).
Lee Rudolph |
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| Lee Rudolph |
| Posted: Sun Dec 21, 2003 7:31 am |
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nulldev00@aol.com (Edward Green) writes:
[quote:ce845e2a19]When physicists speak of "momentum as the generator of spatial
translations", which mathematical sense of "generator" are they
alluding to?
[/quote:ce845e2a19]
The "infinitesimal generator" sense. The sense in which an
element X of a Lie algebra A "generates" (via the exponential
map) a 1-parameter subgroup of a Lie group G whose Lie algebra
is A; more generally (since A can be identified with, for example,
the set of left-invariant vectorfields on G), the sense in which a
vectorfield X on a manifold M "generates" (at least the germ of)
a "flow" on M, i.e., a 1-parameter group of diffeomorphisms of M.
Concretely: consider differentiation d/dt as a linear operator D
on (for instance) the vectorspace V of real polynomials P(t) in a
single variable; form the formal exponential e^D, which takes P
to P+P'+(1/2!)P''+(1/3!)P'''+ ...; luck out (as you do in the
case of V), or work hard, and have e^D converge (in some sense);
compare (e^D)P(t) to P(t+1); gasp in awe.
Lee Rudolph |
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| Leonard Blackburn |
| Posted: Sun Dec 21, 2003 12:00 pm |
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lrudolph@panix.com (Lee Rudolph) wrote in message news:<bs2jb8$21h$1@panix2.panix.com>...
[quote:b588c0580b]blackbur@math.umn.edu (Leonard Blackburn) writes:
nulldev00@aol.com (Edward Green) wrote in message news:<2a0cceff.0312200601.4d6cadbd@posting.google.com>...
The concept of "generator" is popular in physics and mathematics.
Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity...
I believe the unifying concept is the theory of inductive definability.
I believe you believe wrongly (but if you're coming from sci.physics,
who knows?).
[/quote:b588c0580b]
Maybe.
Let (G,*) be a group and let b_1, ..., b_k be fixed members of G.
Let phi(x,S) be the following statement:
"x = b_1 or x = b_2 or ... or x = b_k or
(E y)(E z)[y in S & z in S & x = y*z^{-1}]."
Now define by induction the sets T_n as follows:
x in T_n iff phi(x, Union_{j < n} T_j).
Then the subgroup of G generated by b_1, ..., b_k is
Union_{n} T_n.
So, we've inductively defined the subgroup generated by the given elements.
We can do this with other structures too. This method of definition is
more enlightening than the other "top-down" approach.
-Leonard |
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| Lee Rudolph |
| Posted: Sun Dec 21, 2003 12:16 pm |
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blackbur@math.umn.edu (Leonard Blackburn) writes:
[quote:abb3ff8875]lrudolph@panix.com (Lee Rudolph) wrote in message news:<bs2jb8$21h$1@panix2.panix.com>...
blackbur@math.umn.edu (Leonard Blackburn) writes:
I believe the unifying concept is the theory of inductive definability.
I believe you believe wrongly (but if you're coming from sci.physics,
who knows?).
Maybe.
Let (G,*) be a group and let b_1, ..., b_k be fixed members of G.
Let phi(x,S) be the following statement:
"x = b_1 or x = b_2 or ... or x = b_k or
(E y)(E z)[y in S & z in S & x = y*z^{-1}]."
Now define by induction the sets T_n as follows:
x in T_n iff phi(x, Union_{j < n} T_j).
Then the subgroup of G generated by b_1, ..., b_k is
Union_{n} T_n.
So, we've inductively defined the subgroup generated by the given elements.
We can do this with other structures too. This method of definition is
more enlightening than the other "top-down" approach.
[/quote:abb3ff8875]
I don't see that it's all that much "more enlightening". (1) It
has (what seems to me to be) a specious air of finitude, unless indeed
you intend (as you may) to let k and n be possibly infinite (indeed,
possibly uncountable) ordinals, and simultaneously intend "induction"
to cover "transfinite induction". (2) Even restricting yourself to,
say, finitely generatable countable groups, unless in a particular
case you have reason to know otherwise, the actual growth of the
subsets T_n is unknown (and maybe unknowable)--what you have done,
in effect, is to name the elements of "the subgroup of G generated
by b_1, ..., b_k" by elements of a free group of rank k; but you
haven't (and maybe can't have) told anyone which names are aliases
for the same element--which (to me) extinguishes the purported
enlightenment pretty completely.
Lee Rudolph |
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| Ken Pledger |
| Posted: Sun Dec 21, 2003 4:07 pm |
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In article <2a0cceff.0312200601.4d6cadbd@posting.google.com>,
nulldev00@aol.com (Edward Green) wrote:
[quote:88a0b3734b]The concept of "generator" is popular in physics and mathematics.
Is there a formal unifying concept of "generators", one of several
loosely related similar concepts, or a chance verbal similarity...
[/quote:88a0b3734b]
In mathematics, yes, definitely. It involves the general theory
of closure operations introduced by E. H. Moore in 1910. A couple of
references are:
P. M. Cohn, "Universal Algebra," section II.1 "Closure Systems";
Garrett Birkhoff, "Lattice Theory," the chapter on Complete Lattices.
A closure system on a set A is a subset C of the power set P(A),
such that
(for every subset D of C)(the intersection of all members of D is in C).
This includes the case where D is empty, meaning that A itself is in C.
(An aside. If the above second-order condition causes fluttering
in the logicians' dove-cote, let them flutter! Good mathematics doesn't
have to be first-order.)
Examples are the set C of all subgroups of a group A, and the set
C of all closed subspaces of a topological space A.
Each closure system on A has a corresponding closure operation on
P(A). It's commonly written
X -> (X with an overbar),
but unfortunately ascii presses me to use something like Cohn's
X -> J(X).
A closure system C gives the corresponding closure operation J by the
formula
J(X) = the intersection of all members of C which contain X.
In the two examples above, J(X) is the subgroup generated (N.B.
that word!) by X, or the topological closure of X.
A closure operation J can be defined in its own right as a mapping
from P(A) into itself, satisfying the extensive, isotone and idempotent
laws: (for every X and Y in P(A))
(X is a subset of Y) implies (J(X) is a subset of J(Y)),
X is a subset of J(X),
and J(J(X)) = J(X).
Those subsets of A which are invariant under J form a closure
system C as above, so you can approach the theory from either point of
view.
I like to think of this closure theory (fancifully) as the topmost
part of the foundations of mathematics, where the great branching of the
mathematical tree into continuous and discrete first occurs. That's
because you can specialize closure in the following two significant ways.
The closed subsets of a topology on A form a closure system C
satisfying:
(for every X and Y in C)(the union of X and Y is in C),
and the empty set is in C.
On the other hand, the subalgebras of an algebra A form a closure
system satisfying an ascending chain condition, which means that the
corresponding closure operation J satisfies:
(for every X in P(A))(for every a in J(X))(there is a finite subset F
of X such that a is in J(F)).
Thinking about the group case may help you to understand that. In
the subgroup J(X) generated by a subset X of a group, any single element
a needs only finitely many members of X to express it.
So there you have it! A mathematical object _generated_ by a
subset X usually means J(X) for some relevant closure operation J. The
_generators_ of J(X) are the elements of X.
Ken Pledger. |
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| Lee Rudolph |
| Posted: Sun Dec 21, 2003 4:13 pm |
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Ken Pledger <Ken.Pledger@vuw.ac.nz> writes:
[quote:d87cf4a05c]An aside. If the above second-order condition causes fluttering
in the logicians' dove-cote, let them flutter!
[/quote:d87cf4a05c]
Sort of setting a category among the pigeon-holes, are you?
Lee Rudolph |
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