This section may seem more technical than the previous, but we believe that it is important to understand the meaning of parents and coercion in order to use rings and other algebraic structures in Sage effectively and efficiently.
Note that we try to explain notions, but we do not show here how to implement them. An implementation-oriented tutorial is available as a Sage worksheet.
If one wants to implement a ring in Python, a first approximation is to create a class for the elements X of that ring and provide it with the required double underscore methods such as __add__, __sub__, __mul__, of course making sure that the ring axioms hold.
As Python is a strongly typed (yet dynamically typed) language, one might, at least at first, expect that one implements one Python class for each ring. After all, Python contains one type <int> for the integers, one type <float> for the reals, and so on. But that approach must soon fail: There are infinitely many rings, and one can not implement infinitely many classes.
Instead, one may create a hierarchy of classes designed to implement elements of ubiquitous algebraic structures, such as groups, rings, skew fields, commutative rings, fields, algebras, and so on.
But that means that elements of fairly different rings can have the same type.
sage: P.<x,y> = GF(3) sage: Q.<a,b> = GF(4,'z') sage: type(x)==type(a) True
On the other hand, one could also have different Python classes providing different implementations of the same mathematical structure (e.g., dense matrices versus sparse matrices)
sage: P.<a> = PolynomialRing(ZZ) sage: Q.<b> = PolynomialRing(ZZ, sparse=True) sage: R.<c> = PolynomialRing(ZZ, implementation='NTL') sage: type(a); type(b); type(c) <type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'> <class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse'> <type 'sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl'>
That poses two problems: On the one hand, if one has elements that are two instances of the same class, then one may expect that their __add__ method will allow to add them; but one does not want that, if the elements belong to very different rings. On the other hand, if one has elements belonging to different implementations of the same ring, then one wants to add them, but that is not straight forward if they belong to different Python classes.
The solution to these problems is called “coercion” and will be explained below.
However, it is essential that each element knows what it is element of. That is available by the method parent():
sage: a.parent(); b.parent(); c.parent() Univariate Polynomial Ring in a over Integer Ring Sparse Univariate Polynomial Ring in b over Integer Ring Univariate Polynomial Ring in c over Integer Ring (using NTL)
Similar to the hierarchy of Python classes addressed to elements of algebraic structures, Sage also provides classes for the algebraic structures that contain these elements. Structures containing elements are called “parent structures” in Sage, and there is a base class for them. Roughly parallel to the hierarchy of mathematical notions, one has a hierarchy of classes, namely for sets, rings, fields, and so on:
sage: isinstance(QQ,Field) True sage: isinstance(QQ, Ring) True sage: isinstance(ZZ,Field) False sage: isinstance(ZZ, Ring) True
In algebra, objects sharing the same kind of algebraic structures are collected in so-called “categories”. So, there is a rough analogy between the class hierarchy in Sage and the hierarchy of categories. However, this analogy of Python classes and categories shouldn’t be stressed too much. After all, mathematical categories are implemented in Sage as well:
sage: Rings() Category of rings sage: ZZ.category() Category of euclidean domains sage: ZZ.category().is_subcategory(Rings()) True sage: ZZ in Rings() True sage: ZZ in Fields() False sage: QQ in Fields() True
While Sage’s class hierarchy is centered at implementation details, Sage’s category framework is more centered on mathematical structure. It is possible to implement generic methods and tests independent of a specific implementation in the categories.
Parent structures in Sage are supposed to be unique Python objects. For example, once a polynomial ring over a certain base ring and with a certain list of generators is created, the result is cached:
sage: RR['x','y'] is RR['x','y'] True
The type RingElement should not be confused with the mathematical notion of a ring element; for practical reasons, sometimes an object is an instance of RingElement although it does not belong to a ring:
sage: M = Matrix(ZZ,2,3); M [0 0 0] [0 0 0] sage: isinstance(M, RingElement) True
While parents are unique, equal elements of a parent in Sage are not necessarily identical. This is in contrast to the behaviour of Python for some (albeit not all) integers:
sage: int(1) is int(1) # Python int True sage: int(-15) is int(-15) False sage: 1 is 1 # Sage Integer False
It is important to observe that elements of different rings are in general not distinguished by their type, but by their parent:
sage: a = GF(2)(1) sage: b = GF(5)(1) sage: type(a) is type(b) True sage: parent(a) Finite Field of size 2 sage: parent(b) Finite Field of size 5
Hence, from an algebraic point of view, the parent of an element is more important than its type.
In some cases it is possible to convert an element of one parent structure into an element of a different parent structure. Such conversion can either be explicit or implicit (this is called coercion).
The reader may know the notions type conversion and type coercion from, e.g., the C programming language. There are notions of conversion and coercion in Sage as well. But the notions in Sage are centered on parents, not on types. So, please don’t confuse type conversion in C with conversion in Sage!
We give here a rather brief account. For a detailed description and for information on the implementation, we refer to the section on coercion in the reference manual and to the tutorial worksheet.
There are two extremal positions concerning the possibility of doing arithmetic with elements of different rings:
Sage favours a compromise. If P1 and P2 are parent structures and p1 is an element of P1, then the user may explicitly ask for an interpretation of p1 in P2. This may not be meaningful in all cases or not be defined for all elements of P1, and it is up to the user to ensure that it makes sense. We refer to this as conversion:
sage: a = GF(2)(1) sage: b = GF(5)(1) sage: GF(5)(a) == b True sage: GF(2)(b) == a True
However, an implicit (or automatic) conversion will only happen if this can be done thoroughly and consistently. Mathematical rigour is essential at that point.
Such an implicit conversion is called coercion. If coercion is defined, then it must coincide with conversion. Two conditions must be satisfied for a coercion to be defined:
So, although it is possible to convert each element of GF(2) into GF(5), there is no coercion, since there is no ring homomorphism between GF(2) and GF(5).
The second aspect - consistency - is a bit more difficult to explain. We illustrate it with multivariate polynomial rings. In applications, it certainly makes most sense to have name preserving coercions. So, we have:
sage: R1.<x,y> = ZZ sage: R2 = ZZ['y','x'] sage: R2.has_coerce_map_from(R1) True sage: R2(x) x sage: R2(y) y
If there is no name preserving ring homomorphism, coercion is not defined. However, conversion may still be possible, namely by mapping ring generators according to their position in the list of generators:
sage: R3 = ZZ['z','x'] sage: R3.has_coerce_map_from(R1) False sage: R3(x) z sage: R3(y) x
But such position preserving conversions do not qualify as coercion: By composing a name preserving map from ZZ['x','y'] to ZZ['y','x'] with a position preserving map from ZZ['y','x'] to ZZ['a','b'], a map would result that is neither name preserving nor position preserving, in violation to consistency.
If there is a coercion, it will be used to compare elements of different rings or to do arithmetic. This is often convenient, but the user should be aware that extending the ==-relation across the borders of different parents may easily result in overdoing it. For example, while == is supposed to be an equivalence relation on the elements of one ring, this is not necessarily the case if different rings are involved. For example, 1 in ZZ and in a finite field are considered equal, since there is a canonical coercion from the integers to any finite field. However, in general there is no coercion between two different finite fields. Therefore we have
sage: GF(5)(1) == 1 True sage: 1 == GF(2)(1) True sage: GF(5)(1) == GF(2)(1) False sage: GF(5)(1) != GF(2)(1) True
Similarly, we have
sage: R3(R1.1) == R3.1 True sage: R1.1 == R3.1 False sage: R1.1 != R3.1 True
Another consequence of the consistency condition is that coercions can only go from exact rings (e.g., the rationals QQ) to inexact rings (e.g., real numbers with a fixed precision RR), but not the other way around. The reason is that the composition of the coercion from QQ to RR with a conversion from RR to QQ is supposed to be the identity on QQ. But this is impossible, since some distinct rational numbers may very well be treated equal in RR, as in the following example:
sage: RR(1/10^200+1/10^100) == RR(1/10^100) True sage: 1/10^200+1/10^100 == 1/10^100 False
When comparing elements of two parents P1 and P2, it is possible that there is no coercion between the two rings, but there is a canonical choice of a parent P3 so that both P1 and P2 coerce into P3. In this case, coercion will take place as well. A typical use case is the sum of a rational number and a polynomial with integer coefficients, yielding a polynomial with rational coefficients:
sage: P1.<x> = ZZ sage: p = 2*x+3 sage: q = 1/2 sage: parent(p) Univariate Polynomial Ring in x over Integer Ring sage: parent(p+q) Univariate Polynomial Ring in x over Rational Field
Note that in principle the result would also make sense in the fraction field of ZZ['x']. However, Sage tries to choose a canonical common parent that seems to be most natural (QQ['x'] in our example). If several potential common parents seem equally natural, Sage will not pick one of them at random, in order to have a reliable result. The mechanisms which that choice is based upon is explained in the tutorial worksheet
No coercion into a common parent will take place in the following example:
sage: R.<x> = QQ sage: S.<y> = QQ sage: x+y Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for '+': 'Univariate Polynomial Ring in x over Rational Field' and 'Univariate Polynomial Ring in y over Rational Field'
The reason is that Sage would not choose one of the potential candidates QQ['x']['y'], QQ['y']['x'], QQ['x','y'] or QQ['y','x'], because all of these four pairwise different structures seem natural common parents, and there is no apparent canonical choice.