## 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 cannonical 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

1. If A is B, call a._add_(b)

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

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

4. 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 = \QQ, S = \QQ[x] or R = Gal(S/\QQ) 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.

• 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.

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:
1. If X is R, return x (*)

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

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

4. Return R._element_constructor_(x) (**)

Keywords and extra arguments are passed on. The result of all this logic is cached.
• (*) 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_.

• 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 a Parent 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 Z and our options for Z are \Frac(\Z[x]) or \Frac(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 \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