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7.9.2 Monomial orderings on free algebras
We provide many types of orderings for non-commutative Groebner bases up to a degree (length) bound.
In general it is not clear, whether a given generating set has a finite Groebner
bases with respect to some ordering.
Let
328#328 = {
304#304,...,
305#305} be a set of symbols. A total ordering < on the free monoid
329#329 with
296#296 as the neutral element is called a monomial ordering if
- it is a well-ordering, i.e., every non empty subset has a least element with respect to <, and
- it is compatible with multiplication, that is
330#330 implies
331#331 for all
332#332,
333#333,
4#4 and
46#46 in
329#329.
Note that the latter implies
334#334 for all
297#297 in
329#329.
The left lexicographical ordering on
329#329 with
335#335...
336#336 is defined as follows:
For arbitrary
4#4,
46#46 in
329#329 we say that
337#337, if
-
338#338 or
-
339#339 and 340#340 holds.
Note: left lex is not a monomial ordering, though it is a natural choice to break ties
after, say, comparing elements by the total degree.
In a similar manner one can define the right lexicographical ordering.
On the monoid
341#341define the weight homomorphism
342#342, uniquely determined by
343#343 in
344#344for
345#345.
As a special case, define the length
len:
346#346 by
347#347 for
345#345.
For any ordering << on
329#329 and any weight
342#342 define an ordering
228#228, called the
348#348-weight extension of
349#349 as follows:
For arbitrary
4#4,
46#46 in
329#329 we say that
337#337 if
-
350#350 or
-
351#351 and
352#352 holds.
An ordering < on
329#329 eliminates a certain subset
353#353 if for all
354#354 one has
355#355.
In a ring declaration, LETTERPLACE supports the following monomial orderings.
We illustrate each of the available choices by an example on the free monoid
302#302,
356#356,
357#357,
where we order the monomials
358#358,
359#359,
360#360,
361#361,
362#362,
363#363,
364#364,
365#365,
304#304,
356#356 and
366#366 correspondingly.
- `
dp '
- The degree right lexicographical ordering is the length-weight extension of the right lexicographical ordering.
With respect to the ordering `dp', the test monomials are ordered as follows:
367#367
- `
Dp '
- The degree left lexicographical ordering is the length-weight extension of the left lexicographical ordering.
With respect to the ordering `Dp', the test monomials are ordered as follows:
368#368
- `
Wp(w) for intvec w'
- The weighted degree left lexicographical ordering is the
348#348-weight extension of the left lexicographical ordering with weight
342#342 uniquely determined by strict positive
369#369.
With respect to the ordering `Wp(1, 2, 1)', the test monomials are ordered as follows:
370#370
- `
lp '
- Let 371#371 be weights uniquely determined by
372#372 for
373#373 where 374#374 denotes the Kronecker delta. Let 375#375 be the 376#376-weight extension of the left lexicographical ordering on
377#377 and inductively 378#378 be the 371#371-weight extension of 379#379 for all 380#380.
The monomial ordering lp corresponds to 381#381 and eliminates 382#382 for all 383#383.
We refer to it as to left elimination ordering.
The monomial ordering `lp' corresponds to
381#381 and eliminates {
304#304,...,
384#384} for all
296#296<=
55#55<
17#17.
We refer to it as to left elimination ordering.
With respect to the ordering `lp', the test monomials are ordered as follows:
385#385
- `
rp '
- Let 371#371 be weights uniquely determined by
372#372 for
373#373 where 374#374 denotes the Kronecker delta. Let 381#381 be the 386#386-weight extension of the left lexicographical ordering on
377#377 and inductively 378#378 be the 371#371-weight extension of 387#387 for all 388#388.
The monomial ordering rp corresponds to 375#375 and eliminates
389#389 for all 390#390.
We refer to it as to right elimination ordering.
The monomial ordering `rp' corresponds to
375#375 and eliminates {
384#384,...,
305#305} for all
391#391. We refer to it as to right elimination ordering.
With respect to the ordering `rp', the test monomials are ordered as follows:
392#392
- `
(a(v), ordering) for intvec v'
- For weight
393#393 determined by
394#394 with 395#395 and monomial ordering 396#396 on
377#377, the 333#333-weight extension of 396#396 corresponds to (a(v), o). As a choice for 396#396 there are currently two options implemented, which are dp and Dp. Notice that this ordering eliminates
397#397.
With respect to the ordering `( a(1, 0, 0), Dp)', the test monomials are ordered as follows:
398#398
With ordering `( a(1, 1, 0), Dp)' one obtains:
399#399
The examples are generated by the following code but with customized orderings denoted above.
| LIB "freegb.lib";
ring r = 0, (x1,x2,x3),Dp; // variate ordering here
ring R = freeAlgebra(r, 4);
poly wr = x1*x1*x1+x3*x3*x3+x1*x2*x3+x3*x2*x1+x2*x2+x2*x3+x1*x3+x3*x1+x1+x2+x3;
wr; // polynomial will be automatically ordered according to the ordering on R
==> x1*x1*x1+x1*x2*x3+x3*x2*x1+x3*x3*x3+x1*x3+x2*x2+x2*x3+x3*x1+x1+x2+x3
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