monomial orders for free modules

In Macaulay2, each free module $F = R^s$ over a ring $R$ has a basis of unit column vectors $F_0, F_1, ..., F_(s-1)$. The monomials of $F$ are the elements $m F_i$, where $m$ is a monomial of the ring $R$. In Macaulay2, orders on the monomials of $F$ are used for computing Gröbner bases and syzygies, and also to determine the initial, or lead term of elements of $F$.

The ring $R$ comes equipped with a total order on the monomials of $R$. A total order on the monomials of $F$ is called compatible (with the order on $R$), if $m F_i > n F_i$ (in $F$) whenever $m > n$ (in $R$). There are many types of compatible orders, but several stand out: term over position up (the default in Macaulay2), term over position down, position up over term, position down over term, and Schreyer orders.

term over position up: $m F_i > n F_j$ iff $m>n$ or $m==n$ and $i>j$

term over position down: $m F_i > n F_j$ iff $m>n$ or $m==n$ and $i<j$

position up over term: $m F_i > n F_j$ iff $i>j$ or $i==j$ and $m>n$

position down over term: $m F_i > n F_j$ iff $i<j$ or $i==j$ and $m>n$

Induced monomial orders are another class of important orders on F, see Schreyer orders for their definition and use in Macaulay2.

In Macaulay2, free modules come equipped with a compatible order. The default order is: term over position up. This is called Position=>Up. In the following example, the lead term is $a F_1$, since $a > b$.

 i1 : R = ZZ[a..d]; i2 : F = R^3 3 o2 = R o2 : R-module, free i3 : f = b*F_0 + a*F_1 o3 = | b | | a | | 0 | 3 o3 : R i4 : leadTerm f o4 = | 0 | | a | | 0 | 3 o4 : R
This is the same as giving the monomial order as:
 i5 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Up}]; i6 : F = R^3 3 o6 = R o6 : R-module, free i7 : leadTerm(a*F_0 + a*F_1) o7 = | 0 | | a | | 0 | 3 o7 : R
Giving Position=>Down instead switches the test above to i < j. In this case the monomial order on F is: m*F_i > n*F_j if m>n or m==n and i<j.
 i8 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Down}]; i9 : F = R^3 3 o9 = R o9 : R-module, free i10 : leadTerm(a*F_0 + a*F_1) o10 = | a | | 0 | | 0 | 3 o10 : R
If one gives Position=>Up or Position=>Down earlier, then the position will be taken into account earlier. For example
 i11 : R = ZZ[a..d, MonomialOrder => {GRevLex => 2, Position => Down, GRevLex => 2}]; i12 : F = R^3 3 o12 = R o12 : R-module, free i13 : leadTerm(a*F_0 + a*F_1) o13 = | a | | 0 | | 0 | 3 o13 : R i14 : leadTerm(b*F_0 + c^4*F_1) o14 = | b | | 0 | | 0 | 3 o14 : R i15 : leadTerm(c*F_0 + d^2*F_1) o15 = | c | | 0 | | 0 | 3 o15 : R
If one wants Position over Term (POT), place the Position element first
 i16 : R = ZZ[a..d, MonomialOrder => {Position => Down}]; i17 : F = R^3 3 o17 = R o17 : R-module, free i18 : leadTerm(a*F_0 + a*F_1) o18 = | a | | 0 | | 0 | 3 o18 : R i19 : leadTerm(b*F_0 + c^4*F_1) o19 = | b | | 0 | | 0 | 3 o19 : R i20 : leadTerm(c*F_0 + d^2*F_1) o20 = | c | | 0 | | 0 | 3 o20 : R