Contents

## Preliminaries

### What is coercion all about?

*The primary goal of coercion is to be able to transparently do arithmetic, comparisons, etc. between elements of distinct sets.*

As a concrete example, when one writes 1 + 1/2 one wants to perform arithmetic on the operands as rational numbers, despite the left being an integer. This makes sense given the obvious and natural inclusion of the integers into the rational numbers. The goal of the coercion system is to facilitate this (and more complicated arithmetic) without having to explicitly map everything over into the same domain, and at the same time being strict enough to not resolve ambiguity or accept nonsense. Here are some examples:

sage: 1 + 1/2 3/2 sage: R.<x,y> = ZZ[] sage: R Multivariate Polynomial Ring in x, y over Integer Ring sage: parent(x) Multivariate Polynomial Ring in x, y over Integer Ring sage: parent(1/3) Rational Field sage: x+1/3 x + 1/3 sage: parent(x+1/3) Multivariate Polynomial Ring in x, y over Rational Field sage: GF(5)(1) + CC(I) Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for '+': 'Finite Field of size 5' and 'Complex Field with 53 bits of precision'

### Parents and Elements

Parents are objects in concrete categories, and Elements are their members. Parents are first-class objects. Most things in Sage are either parents or have a parent. Typically whenever one sees the word *Parent* one can think *Set*. Here are some examples:

sage: parent(1) Integer Ring sage: parent(1) is ZZ True sage: ZZ Integer Ring sage: parent(1.50000000000000000000000000000000000) Real Field with 123 bits of precision sage: parent(x) Symbolic Ring sage: x^sin(x) x^sin(x) sage: R.<t> = Qp(5)[] sage: f = t^3-5; f (1 + O(5^20))*t^3 + (4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + 4*5^12 + 4*5^13 + 4*5^14 + 4*5^15 + 4*5^16 + 4*5^17 + 4*5^18 + 4*5^19 + 4*5^20 + O(5^21)) sage: parent(f) Univariate Polynomial Ring in t over 5-adic Field with capped relative precision 20 sage: f = EllipticCurve('37a').lseries().taylor_series(10); f 0.997997869801216 + 0.00140712894524925*z - 0.000498127610960097*z^2 + 0.000118835596665956*z^3 - 0.0000215906522442707*z^4 + (3.20363155418419e-6)*z^5 + O(z^6) sage: parent(f) Power Series Ring in z over Complex Field with 53 bits of precision

There is an important distinction between Parents and types

sage: a = GF(5).random_element() sage: b = GF(7).random_element() sage: type(a) <type 'sage.rings.integer_mod.IntegerMod_int'> sage: type(b) <type 'sage.rings.integer_mod.IntegerMod_int'> sage: type(a) == type(b) True sage: parent(a) Finite Field of size 5 sage: parent(a) == parent(b) False

However, non-sage objects don't really have parents, but we still want to be able to reason with them, so their type is used instead.

sage: a = int(10) sage: parent(a) <type 'int'>

In fact, under the hood, a special kind of parent "The set of all Python objects of type T" is used in these cases.

Note that parents are **not** always as tight as possible.

sage: parent(1/2) Rational Field sage: parent(2/1) Rational Field

### Maps between Parents

Many parents come with maps to and from other parents.

Sage makes a distinction between being able to **convert** between various parents, and **coerce** between them. Conversion is explicit and tries to make sense of an object in the target domain if at all possible. It is invoked by calling

sage: ZZ(5) 5 sage: ZZ(10/5) 2 sage: QQ(10) 10 sage: parent(QQ(10)) Rational Field sage: a = GF(5)(2); a 2 sage: parent(a) Finite Field of size 5 sage: parent(ZZ(a)) Integer Ring sage: GF(71)(1/5) 57 sage: ZZ(1/2) ... TypeError: no conversion of this rational to integer

Conversions need not be canonical (they may for example involve a choice of lift) or even make sense mathematically (e.g. constructions of some kind).

sage: ZZ("123") 123 sage: ZZ(GF(5)(14)) 4 sage: ZZ['x']([4,3,2,1]) x^3 + 2*x^2 + 3*x + 4 sage: a = Qp(5, 10)(1/3); a 2 + 3*5 + 5^2 + 3*5^3 + 5^4 + 3*5^5 + 5^6 + 3*5^7 + 5^8 + 3*5^9 + O(5^10) sage: ZZ(a) 6510417

On the other hand, Sage has the notion of a **coercion**, which is a canonical morphism (occasionally up to a conventional choice made by developers) between parents. A coercion from one parent to another **must** be defined on the whole domain, and always succeeds. As it may be invoked implicitly, it should be obvious and natural (in both the mathematically rigorous and colloquial sense of the word). Up to inescapable rounding issues that arise with inexact representations, these coercion morphisms should all commute.

They can be discovered via the `has_coerce_map_from` method, and if needed explicitly invoked with the `coerce` method.

sage: QQ.has_coerce_map_from(ZZ) True sage: QQ.has_coerce_map_from(RR) False sage: ZZ['x'].has_coerce_map_from(QQ) False sage: ZZ['x'].has_coerce_map_from(ZZ) True sage: ZZ['x'].coerce(5) 5 sage: ZZ['x'].coerce(5).parent() Univariate Polynomial Ring in x over Integer Ring sage: ZZ['x'].coerce(5/1) Traceback (most recent call last): ... TypeError: no canonical coercion from Rational Field to Univariate Polynomial Ring in x over Integer Ring

## Basic Arithmetic Rules

Suppose we want to add two element, a and b, whose parents are A and B respectively. When we type `a+b` then

If A

`is`B, call a._add_(b)If there is a coercion \phi: B \rightarrow A, call a._add_(\phi(b))

If there is a coercion \phi: A \rightarrow B, call \phi(a)._add_(b)

Look for Z such that there is a coercion \phi_A: A \rightarrow Z and \phi_B: B \rightarrow Z, call \phi_A(a)._add_(\phi_B(b))

The same rules are used for subtraction, multiplication, and division. This logic is embedded in a coercion model object, which can be obtained and queried.

sage: parent(1 + 1/2) Rational Field sage: cm = sage.structure.element.get_coercion_model() <sage.structure.coerce.CoercionModel_cache_maps object at 0x2f65960> sage: cm.explain(ZZ, QQ) Coercion on left operand via Natural morphism: From: Integer Ring To: Rational Field Arithmetic performed after coercions. Result lives in Rational Field Rational Field sage: cm.explain(ZZ['x','y'], QQ['x']) Coercion on left operand via Call morphism: From: Multivariate Polynomial Ring in x, y over Integer Ring To: Multivariate Polynomial Ring in x, y over Rational Field Coercion on right operand via Call morphism: From: Univariate Polynomial Ring in x over Rational Field To: Multivariate Polynomial Ring in x, y over Rational Field Arithmetic performed after coercions. Result lives in Multivariate Polynomial Ring in x, y over Rational Field Multivariate Polynomial Ring in x, y over Rational Field

The coercion model can be used directly for any binary operation (callable taking two arguments).

sage: cm.bin_op(77, 9, gcd) 1

There are also **actions** in the sense that a field K acts on a module over K, or a permutation group acts on a set. These are discovered between steps 1 and 2 above.

sage: cm.explain(ZZ['x'], ZZ, operator.mul) Action discovered. Right scalar multiplication by Integer Ring on Univariate Polynomial Ring in x over Integer Ring Result lives in Univariate Polynomial Ring in x over Integer Ring Univariate Polynomial Ring in x over Integer Ring sage: cm.explain(ZZ['x'], ZZ, operator.div) Action discovered. Right inverse action by Rational Field on Univariate Polynomial Ring in x over Integer Ring with precomposition on right by Natural morphism: From: Integer Ring To: Rational Field Result lives in Univariate Polynomial Ring in x over Rational Field Univariate Polynomial Ring in x over Rational Field sage: f = QQ.coerce_map_from(ZZ) sage: f(3).parent() Rational Field sage: QQ.coerce_map_from(int) Native morphism: From: Set of Python objects of type 'int' To: Rational Field sage: QQ.has_coerce_map_from(RR) False sage: QQ['x'].get_action(QQ) Right scalar multiplication by Rational Field on Univariate Polynomial Ring in x over Rational Field sage: (QQ^2).get_action(QQ) Right scalar multiplication by Rational Field on Vector space of dimension 2 over Rational Field sage: QQ['x'].get_action(RR) Right scalar multiplication by Real Field with 53 bits of precision on Univariate Polynomial Ring in x over Rational Field

## How to Implement

### Methods to implement

Arithmetic on Elements

`_add_, _sub_, _mul_, _div_`This is where the binary arithmetic operators should be implemented. Unlike Python's

`__add__`, both operands are*guaranteed*to have the same Parent at this point.

Coercion for Parents

`_coerce_map_from_`Given two parents R and S,

`R._coerce_map_from_(S)`is called to determine if there is a coercion \phi: S \rightarrow R. Note that the function is called on the potential codomain. To indicate that there is no coercion from S to R (self), return`False`or`None`. This is the default behavior. If there is a coercion, return`True`(in which case an morphism using`R._element_constructor_`will be created) or an actual`Morphism`object with S as the domain and R as the codomain.

Actions for Parents

`_get_action_`or`_rmul_, _lmul_, _r_action_, _l_action_`Suppose one wants R to act on S. Some examples of this could be R = \mathbb{Q}, S = \mathbb{Q}[x] or R = Gal(S/\mathbb{Q}) where S is a number field. There are several ways to implement this:

If R is the base of S (as in the first example), simply implement

`_rmul_`and/or`_lmul_`on the Elements of S.In this case

`r * s`gets handled as`s._rmul_(r)`and`s * r`as`s._lmul_(r)`. The argument to`_rmul_`and`_lmul_`are*guaranteed*to be Elements of the base of S (with coercion happening beforehand if necessary).

If R acts on S, one can alternatively define the methods

`_r_action_`and/or`_l_action_`on the Elements of R.There is no constraint on the type or parents of objects passed to these methods; raise a

`TypeError`or`ValueError`if the wrong kind of object is passed in to indicate the action is not appropriate here.**Important**Due to potential confusion, it has been decided to rename these to`_act_on_`and`_acted_upon_`

If either R acts on S

`or``S acts on R, one may implement``R._get_action_`to return an actual`Action`object to be used.- This is how non-multiplicative actions must be implemented, and is the most powerful (and completed) way to do things.

Element conversion/construction for Parents: use

`_element_constructor_`*not*`__call__`The

`Parent.__call__`method dispatches to`_element_constructor_`. When someone writes R(x, ...), this is the method that eventually gets called in most cases. See the documentation on the`__call__`method below.

Parents may also call the `self._populate_coercion_lists_` method in their `__init__` functions to pass any callable for use instead of `_element_constructor_`, provide a list of Parents with coercions to self (as an alternative to implementing `_coerce_map_from_`), provide special construction methods (like `_integer_` for ZZ), etc. This also allows one to specify a single coercion embedding *out* of self (whereas the rest of the coercion functions all specify maps *into* self). There is extensive documentation in the docstring of the `_populate_coercion_lists_` method.

### Example

Sometimes a **simple example** is worth a thousand words.

### Provided Methods

`__call__`- This provides a consistent interface for element construction. In particular, it makes sure that conversion always gives the same result as coercion, if a coercion exists. (This used to be violated for some Rings in Sage as the code for conversion and coercion got edited separately.) Let R be a Parent and assume the user types R(x), where x has parent X. Roughly speaking, the following occurs:
If X

`is`R, return x (*)If there is a coercion f: X \rightarrow R, return f(x)

If there is a coercion f: R \rightarrow X, try to return {f^{-1}}(x)

Return

`R._element_constructor_(x)`(**)

- (*) Unless there is a "copy" keyword like R(x, copy=False)
(**) Technically, a generic morphism is created from X to R, which may use magic methods like

`_integer_`or other data provided by`_populate_coercion_lists_`.

- This provides a consistent interface for element construction. In particular, it makes sure that conversion always gives the same result as coercion, if a coercion exists. (This used to be violated for some Rings in Sage as the code for conversion and coercion got edited separately.) Let R be a Parent and assume the user types R(x), where x has parent X. Roughly speaking, the following occurs:
`coerce`- Coerces elements into self, raising a type error if there is no coercion map.

`coerce_map_from, convert_map_from`Returns an actual

`Morphism`object to coerce/convert from another Parent to self. Barring direct construction of elements of R,`R.convert_map_from(S)`will provide a callable Python object which is the fastest way to convert elements of S to elements of R. From Cython, it can be invoked via the cdef`_call_`method.

`has_coerce_map_from`Returns

`True`or`False`depending on whether or not there is a coercion.`R.has_coerce_map_from(S)`is shorthand for`R.coerce_map_from(S) is not None`

`get_action`This will unwind all the

`_rmul_, _lmul_, _r_action_, _l_action_, ...`methods to provide an actual`Action`object, if one exists.

## Discovering new parents

New parents are discovered using an algorithm in sage/category/pushout.py. The fundamental idea is that most Parents in Sage are constructed from simpler objects via various functors. These are accessed via the `construction` method, which returns a (simpler) Parent along with a functor with which one can create self.

sage: CC.construction() (AlgebraicClosureFunctor, Real Field with 53 bits of precision) sage: RR.construction() (CompletionFunctor, Rational Field) sage: QQ.construction() (FractionField, Integer Ring) sage: ZZ.construction() # None sage: Qp(5).construction() (CompletionFunctor, Rational Field) sage: QQ.completion(5, 100) 5-adic Field with capped relative precision 100 sage: c, R = RR.construction() sage: a = CC.construction()[0] sage: a.commutes(c) False sage: RR == c(QQ) True sage: sage.categories.pushout.construction_tower(Frac(CDF['x'])) [(None, Fraction Field of Univariate Polynomial Ring in x over Complex Double Field), (FractionField, Univariate Polynomial Ring in x over Complex Double Field), (Poly[x], Complex Double Field), (AlgebraicClosureFunctor, Real Double Field), (CompletionFunctor, Rational Field), (FractionField, Integer Ring)]

Given Parents R and S, such that there is no coercion either from R to S or from S to R, one can find a common Z with coercions R \rightarrow Z and S \rightarrow Z by considering the sequence of construction functors to get from a common ancestor to both R and S. We then use a *heuristic* algorithm to interleave these constructors in an attempt to arrive at a suitable Z (if one exists). For example:

sage: ZZ['x'].construction() (Poly[x], Integer Ring) sage: QQ.construction() (FractionField, Integer Ring) sage: sage.categories.pushout.pushout(ZZ['x'], QQ) Univariate Polynomial Ring in x over Rational Field sage: sage.categories.pushout.pushout(ZZ['x'], QQ).construction() (Poly[x], Rational Field)

The common ancestor is \mathbb{Z} and our options for Z are Frac(\mathbb{Z}[x]) or Frac(\mathbb{Z})[x]. In Sage we choose the later, treating the fraction field functor as binding "more tightly" than the polynomial functor, as most people agree that \mathbb{Q}[x] is the more natural choice. The same procedure is applied to more complicated Parents, returning a new Parent if one can be unambiguously determined.

sage: sage.categories.pushout.pushout(Frac(ZZ['x,y,z']), QQ['z, t']) Fraction Field of Multivariate Polynomial Ring in x, y, z, t over Rational Field

# Coercion Example

Here is a minimal example of setting up a simple Ring that handles coercion. (It is easy to imagine much more sophisticated and powerful localizations, but that would obscure the main points being made here.)

class Localization(Ring): def __init__(self, primes): """ Localization of $\Z$ away from primes. """ Ring.__init__(self, base=ZZ) self._primes = primes self._populate_coercion_lists_() def _repr_(self): """ How to print self. """ return "%s localized at %s" % (self.base(), self._primes) def _element_constructor_(self, x): """ Make sure x is a valid member of self, and return the constructed element. """ if isinstance(x, LocalizationElement): x = x._x else: x = QQ(x) for p, e in x.denominator().factor(): if p not in self._primes: raise ValueError, "Not integral at %s" % p return LocalizationElement(self, x) def _coerce_map_from_(self, S): """ The only things that coerce into this ring are: - the integer ring - other localizations away from fewer primes """ if S is ZZ: return True elif isinstance(S, Localization): return all(p in self._primes for p in S._primes) class LocalizationElement(RingElement): def __init__(self, parent, x): RingElement.__init__(self, parent) self._value = x # We're just printing out this way to make it easy to see what's going on in the examples. def _repr_(self): return "LocalElt(%s)" % self._value # Now define addition, subtraction, and multiplication of elements. # Note that left and right always have the same parent. def _add_(left, right): return LocalizationElement(left.parent(), left._value + right._value) def _sub_(left, right): return LocalizationElement(left.parent(), left._value - right._value) def _mul_(left, right): return LocalizationElement(left.parent(), left._value * right._value) # The basering was set to ZZ, so c is guaranteed to be in ZZ def _rmul_(self, c): return LocalizationElement(self.parent(), c * self._value) def _lmul_(self, c): return LocalizationElement(self.parent(), self._value * c) That's all there is to it. Now we can test it out: sage: R = Localization([2]); R Integer Ring localized at [2] sage: R(1) LocalElt(1) sage: R(1/2) LocalElt(1/2) sage: R(1/3) Traceback (most recent call last): ... ValueError: Not integral at 3 sage: R.coerce(1) LocalElt(1) sage: R.coerce(1/4) Traceback (click to the left for traceback) ... TypeError: no cannonical coercion from Rational Field to Integer Ring localized at [2] sage: R(1/2) + R(3/4) LocalElt(5/4) sage: R(1/2) + 5 LocalElt(11/2) sage: 5 + R(1/2) LocalElt(11/2) sage: R(1/2) + 1/7 Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for '+': 'Integer Ring localized at [2]' and 'Rational Field' sage: R(3/4) * 7 LocalElt(21/4) sage: R.get_action(ZZ) Right scalar multiplication by Integer Ring on Integer Ring localized at [2] sage: cm = sage.structure.element.get_coercion_model() sage: cm.explain(R, ZZ, operator.add) Coercion on right operand via Conversion map: From: Integer Ring To: Integer Ring localized at [2] Arithmetic performed after coercions. Result lives in Integer Ring localized at [2] Integer Ring localized at [2] sage: cm.explain(R, ZZ, operator.mul) Action discovered. Right scalar multiplication by Integer Ring on Integer Ring localized at [2] Result lives in Integer Ring localized at [2] Integer Ring localized at [2] sage: R6 = Localization([2,3]); R6 Integer Ring localized at [2, 3] sage: R6(1/3) - R(1/2) LocalElt(-1/6) sage: parent(R6(1/3) - R(1/2)) Integer Ring localized at [2, 3] sage: R.has_coerce_map_from(ZZ) True sage: R.coerce_map_from(ZZ) Conversion map: From: Integer Ring To: Integer Ring localized at [2] sage: R6.coerce_map_from(R) Conversion map: From: Integer Ring localized at [2] To: Integer Ring localized at [2, 3] sage: R6.coerce(R(1/2)) LocalElt(1/2) sage: cm.explain(R, R6, operator.mul) Coercion on left operand via Conversion map: From: Integer Ring localized at [2] To: Integer Ring localized at [2, 3] Arithmetic performed after coercions. Result lives in Integer Ring localized at [2, 3] Integer Ring localized at [2, 3]