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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 326#326 = { 302#302,..., 303#303} be a set of symbols. A total ordering < on the free monoid 327#327 with 294#294 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 328#328 implies 329#329 for all 330#330, 331#331, 4#4 and 46#46 in 327#327.
Note that the latter implies 332#332 for all 295#295 in 327#327.

The left lexicographical ordering on 327#327 with 333#333... 334#334 is defined as follows: For arbitrary 4#4, 46#46 in 327#327 we say that 335#335, if

  • 336#336 or

  • 337#337 and 338#338 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 339#339define the weight homomorphism 340#340, uniquely determined by 341#341 in 342#342for 343#343.

As a special case, define the length len: 344#344 by 345#345 for 343#343.

For any ordering << on 327#327 and any weight 340#340 define an ordering 226#226, called the 346#346-weight extension of 347#347 as follows: For arbitrary 4#4, 46#46 in 327#327 we say that 335#335 if

  • 348#348 or
  • 349#349 and 350#350 holds.
An ordering < on 327#327 eliminates a certain subset 351#351 if for all 352#352 one has 353#353.

In a ring declaration, LETTERPLACE supports the following monomial orderings.

We illustrate each of the available choices by an example on the free monoid 300#300, 354#354, 355#355, where we order the monomials

356#356, 357#357, 358#358, 359#359, 360#360, 361#361, 362#362, 363#363, 302#302, 354#354 and 364#364 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:

365#365

`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:

366#366

`Wp(w) for intvec w'
The weighted degree left lexicographical ordering is the 346#346-weight extension of the left lexicographical ordering with weight 340#340 uniquely determined by strict positive 367#367.

With respect to the ordering `Wp(1, 2, 1)', the test monomials are ordered as follows:

368#368

`lp'
Let 369#369 be weights uniquely determined by 370#370 for 371#371 where 372#372 denotes the Kronecker delta. Let 373#373 be the 374#374-weight extension of the left lexicographical ordering on 375#375 and inductively 376#376 be the 369#369-weight extension of 377#377 for all 378#378. The monomial ordering lp corresponds to 379#379 and eliminates 380#380 for all 381#381. We refer to it as to left elimination ordering.

The monomial ordering `lp' corresponds to 379#379 and eliminates { 302#302,..., 382#382} for all 294#294<= 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:

383#383

`rp'
Let 369#369 be weights uniquely determined by 370#370 for 371#371 where 372#372 denotes the Kronecker delta. Let 379#379 be the 384#384-weight extension of the left lexicographical ordering on 375#375 and inductively 376#376 be the 369#369-weight extension of 385#385 for all 386#386. The monomial ordering rp corresponds to 373#373 and eliminates 387#387 for all 388#388. We refer to it as to right elimination ordering.

The monomial ordering `rp' corresponds to 373#373 and eliminates { 382#382,..., 303#303} for all 389#389. We refer to it as to right elimination ordering.

With respect to the ordering `rp', the test monomials are ordered as follows:

390#390

`(a(v), ordering) for intvec v'
For weight 391#391 determined by 392#392 with 393#393 and monomial ordering 394#394 on 375#375, the 331#331-weight extension of 394#394 corresponds to (a(v), o). As a choice for 394#394 there are currently two options implemented, which are dp and Dp. Notice that this ordering eliminates 395#395.

With respect to the ordering `( a(1, 0, 0), Dp)', the test monomials are ordered as follows:

396#396

With ordering `( a(1, 1, 0), Dp)' one obtains:

397#397

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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