# Element Operations ## Arithmetic Operators The usual unary and binary ring operations are available for noncommutative polynomials, noting that multiplication is associative but noncommutative, of course. ### `+ a: AlgFrElt -> AlgFrElt` ### `- a: AlgFrElt -> AlgFrElt` ### `a + b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a - b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a * b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a ^ k: AlgFrElt, RngIntElt -> AlgFrElt` ### `a / b: AlgFrElt, AlgFrElt -> FldFunMElt` ### `a div b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a +:= b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a -:= b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a *:= b: AlgFrElt, AlgFrElt -> AlgFrElt` ### `a div:= b: AlgFrElt, AlgFrElt -> AlgFrElt` ## Equality and Membership ### `a eq b: AlgFrElt, AlgFrElt -> BoolElt` ### `a ne b: AlgFrElt, AlgFrElt -> BoolElt` ### `a in R: AlgFrElt, Rng -> BoolElt` ### `a notin R: AlgFrElt, Rng -> BoolElt` ## Predicates on Algebra Elements ### `IsZero(f): AlgFrElt -> BoolElt` ### `IsOne(f): AlgFrElt -> BoolElt` ### `IsMinusOne(f): AlgFrElt -> BoolElt` ### `IsNilpotent(f): AlgFrElt -> BoolElt` ### `IsIdempotent(f): AlgFrElt -> BoolElt` ### `IsUnit(f): AlgFrElt -> BoolElt` ### `IsZeroDivisor(f): AlgFrElt -> BoolElt` ### `IsRegular(f): AlgFrElt -> BoolElt` ### `IsIrreducible(f): AlgFrElt -> BoolElt` ### `IsPrime(f): AlgFrElt -> BoolElt` ## Coefficients, Monomials, Terms and Degree The functions in this subsection allow one to access noncommutative polynomials. ### `Coefficients(f): AlgFrElt -> [ RngElt ]` Given a noncommutative polynomial $f$ with coefficients in $R$, this function returns a sequence of ‘base’ coefficients, that is, a sequence of elements of $R$ occurring as coefficients of the monomials in $f$. Note that the monomials are ordered, and that the sequence of coefficients corresponds exactly to the sequence of monomials returned by `Monomials(f)`. ### `LeadingCoefficient(f): AlgFrElt -> RngElt` Given a noncommutative polynomial $f$ with coefficients in $R$, this function returns the leading coefficient of $f$ as an element of $R$; this is the coefficient of the leading monomial of $f$, that is, the first among the monomials occurring in $f$ with respect to the ordering of monomials used in $F$. ### `TrailingCoefficient(f): AlgFrElt -> RngElt` Given a noncommutative polynomial $f$ with coefficients in $R$, this function returns the trailing coefficient of $f$ as an element of $R$; this is the coefficient of the trailing monomial of $f$, that is, the last among the monomials occurring in $f$ with respect to the ordering of monomials used in $F$. ### `MonomialCoefficient(f, m): AlgFrElt, AlgFrElt -> RngElt` Given a noncommutative polynomial $f$ and a monomial $m$, this function returns the coefficient with which $m$ occurs in $f$ as an element of $R$. ### `Monomials(f): AlgFrElt -> [ AlgFrElt ]` Given a noncommutative polynomial $f\in F$, this function returns a sequence of the monomials (monoid words) occurring in $f$. Note that the monomials in $F$ are ordered, and that the sequence of monomials corresponds exactly to the sequence of coefficients returned by `Coefficients(f)`. ### `LeadingMonomial(f): AlgFrElt -> AlgFrElt` Given a noncommutative polynomial $f\in F$ this function returns the leading monomial of $f$, that is, the first monomial element of $F$ that occurs in $f$, with respect to the ordering of monomials used in $F$. ### `Terms(f): AlgFrElt -> [ AlgFrElt ]` Given a noncommutative polynomial $f\in F$, this function returns the sequence of (non-zero) terms of $f$ as elements of $F$. The terms are ordered according to the ordering on the monomials in $F$. Consequently the $i$-th element of this sequence of terms will be equal to the product of the $i$-th element of the sequence of coefficients and the $i$-th element of the sequence of monomials. ### `LeadingTerm(f): AlgFrElt -> AlgFrElt` Given a noncommutative polynomial $f\in F$, this function returns the leading term of $f$ as an element of $F$; this is the product of the leading monomial and the leading coefficient that is, the first among the monomial terms occurring in $f$ with respect to the ordering of monomials used in $F$. ### `TrailingTerm(f): AlgFrElt -> RngElt` Given a noncommutative polynomial $f\in F$, this function returns the trailing term of $f$ as an element of $F$; this is the last among the monomial terms occurring in $f$ with respect to the ordering of monomials used in $F$. ### `Length(m): AlgFrElt -> RngIntElt` Given a noncommutative monomial (word) $m$, return the length of $m$, i.e., the number of letters of $m$. Note that this differs from the commutative case, where the number of terms in a polynomial is returned. ### `m[i]: AlgFrElt, RngIntElt -> AlgFrElt` Given a noncommutative monomial (word) $m$ of length $l$, and an integer $i$ with $1\leq i\leq l$, return the $i$-th letter of $m$. ### `TotalDegree(f): AlgFrElt -> RngIntElt` Given a noncommutative polynomial $f$, this function returns the total degree of $f$, which is the maximum of the lengths of all monomials that occur in $f$. If $f$ is the zero polynomial, the return value is $-1$. ### `LeadingTotalDegree(f): AlgFrElt -> RngIntElt` Given a noncommutative polynomial, this function returns the leading total degree of $f$, which is the length of the leading monomial of $f$. ### `Example: Terms (ex-4ecfe7)` In this example we illustrate the above access functions. ```magma > K := RationalField(); > F := FreeAlgebra(K, 3); > f := (3*x*y - 2*y*z)*(4*x - 7*z*y) + 23*x*y*z; > f; -21*x*y*z*y + 14*y*z^2*y + 12*x*y*x + 23*x*y*z - 8*y*z*x > TotalDegree(f); 4 > Coefficients(f); [ -21, 14, 12, 23, -8 ] > Monomials(f); [ x*y*z*y, y*z^2*y, x*y*x, x*y*z, y*z*x ] > Terms(f); [ -21*x*y*z*y, 14*y*z^2*y, 12*x*y*x, 23*x*y*z, -8*y*z*x ] > MonomialCoefficient(f, x*y*z); 23 > LeadingTerm(f); -21*x*y*z*y > LeadingCoefficient(f); -21 > m := Monomials(f)[1]; > m; x*y*z*y > Length(m); 4 > m[1]; x > m[2]; y ``` ## Evaluation ### `Evaluate(f, s): AlgFrElt, [ RngElt ] -> RngElt` ### `Evaluate(f, s): AlgFrElt, < RngElt, ..., RngElt > -> RngElt` Given an element $f$ of a free algebra $F=R\langle x_1, \ldots, x_n\rangle$ and a sequence or tuple $s$ of ring or algebra elements of length $n$, return the value of $f$ at $s$, that is, the value obtained by substituting $x_i=s[i]$. This behaves in the same way as the `hom` constructor above. If the elements of $s$ lie in a ring and can be lifted into the coefficient ring $R$, then the result will be an element of $R$. If the elements of $s$ cannot be lifted to the coefficient ring, then an attempt is made to do a generic evaluation of $f$ at $s$. In this case, the result will be of the same type as the elements of $s$. ### `Example: Terms (ex-88ec97)` In this example we illustrate the above access functions. ```magma > K := RationalField(); > F := FreeAlgebra(K, 3); > g := x*y + y*z; > g; x*y + y*z > Evaluate(g, [1,2,3]); 8 > Parent($1); Rational Field > Evaluate(g, [y,x,z]); x*z + y*x > Parent($1); Finitely presented algebra of rank 3 over Rational Field Non-commutative Graded Lexicographical Order Variables: x, y, z ```