[Cyan]<CedarChest6.0>AlgebraStructures>AlgebraClasses.mesa!1

AlgebraClasses.mesa

Arnon, March 4, 1986 12:02:24 pm PST

DIRECTORY

SafeStorage,

IO,

Atom,

Rope,

Basics;

AlgebraClasses: CEDAR DEFINITIONS

= BEGIN

Structure Type

Structure: TYPE = REF StructureRec;

StructureRec: TYPE = RECORD [

class: StructureClass,

instanceData: REF

];

Class Record Type

ClassFlavor: TYPE = {group, ring, field, vectorSpace, algebra, divisionAlgebra};

StructureClass: TYPE = REF StructureClassRec;

StructureClassRec: TYPE = RECORD [

Structure properties

flavor: ClassFlavor,

printName: PrintNameProc,

characteristic: StructureRankOp ← NIL, -- rings, fields, vector spaces, algebras, divisionAlgebras only

dimension: StructureRankOp ← NIL,-- vector spaces, algebras, divisionAlgebras only

Test whether a given item belongs to this structure (run-time typechecking)

isElementOf: ElementOfProc ← NIL,

I/O

legalFirstChar: LegalFirstCharOp,

read: ReadOp,

fromRope: FromRopeOp,

toRope: ToRopeOp,

write: WriteOp,

Addition (rings, fields, vectorSpaces, algebras, divisionAlgebras only)

add: BinaryOp ← NIL,

negate: UnaryOp ← NIL,

subtract: BinaryOp ← NIL,

zero: NullaryOp ← NIL,

Multiplication (groups, rings, fields, algebras, divisionAlgebras only)

multiply: BinaryOp, -- assumed associative; commutative or noncommutative ok

commutative: BOOL ← TRUE,

invert: UnaryOp ← NIL,

divide: BinaryOp ← NIL,

one: NullaryOp,

Scalar multiplication (vectorSpaces, algebras, divisionAlgebras only)

scalarMultiply: BinaryOp ← NIL, -- this is left multiplication, i.e. scalar * vector. The underlying field of a vector space may be commutative or noncommutative.

Equality testing

equal: EqualityOp,

Ordered structure ops (rings, fields, vectorSpaces, algebras, divisionAlgebras only)

ordered: BOOL ← TRUE,

sign: CompareToZeroOp ← NIL,

abs: UnaryOp ← NIL,

compare: BinaryCompareOp ← NIL,

Some common varieties of rings (and algebras)

integralDomain: BOOL ← FALSE,

gcdDomain: BOOL ← FALSE,

gcd: BinaryOp ← NIL, -- gcdDomains only

euclideanDomain: BOOL ← FALSE,

remainder: BinaryOp ← NIL, -- euclideanDomains only

degree: ElementRankOp ← NIL, -- euclideanDomains only

Some common varieties of fields (and divisionAlgebras)

completeField: BOOL ← FALSE,

realField: BOOL ← FALSE,

realClosedField: BOOL ← FALSE,

algebraicallyClosedField: BOOL ← FALSE,

Additional ops and properties go on property list, for example:

$PolynomialRing

$MatrixRing

$EuclideanDomainMod

$ExtensionField (ground field cannot be an algebraicallyClosedField)

$RealExtensionField (ground field must be a realField, not also a realClosedField)

$QuotientField

$PowerSeriesRing

propList: Atom.PropList ← NIL

];

Structure Operation Types

PrintNameProc: TYPE = PROC [structure: Structure] RETURNS [Rope.ROPE];

Identify the particular structure to the world.

StructureRankOp: TYPE = PROC [structure: Structure] RETURNS [CARDINAL];

For things like ring characteristic (smallest positive integer k such that k*1 = 0; 0 if k infinite) and vector space dimension.

Structure Element Operation Types

ElementOfProc: TYPE = PROC [item: REF, structure: Structure] RETURNS [BOOL];

True if item is an element of the structure.

LegalFirstCharOp: TYPE = PROC [char: CHAR, structure: Structure] RETURNS [BOOL];

True if char is legal first char in an external rep of a structure element.

ReadOp: TYPE = PROC [in: IO.STREAM, structure: Structure ← NIL] RETURNS [out: REF];

FromRopeOp: TYPE = PROC [in: Rope.ROPE, structure: Structure ← NIL] RETURNS [out: REF];

ToRopeOp: TYPE = PROC [in: REF, structure: Structure ← NIL] RETURNS [out: Rope.ROPE];

WriteOp: TYPE = PROC [stream: IO.STREAM, in: REF, structure: Structure ← NIL];

NullaryOp: TYPE = PROC[structure: Structure ← NIL] RETURNS [result: REF];

Note that since no args, no hooks to structure

UnaryOp: TYPE = PROC [arg: REF, structure: Structure ← NIL] RETURNS [result: REF];

BinaryOp: TYPE = PROC [firstArg, secondArg: REF, structure: Structure ← NIL] RETURNS [result: REF];

EqualityOp: TYPE = PROC [firstArg, secondArg: REF, structure: Structure ← NIL] RETURNS [BOOL];

CompareToZeroOp: TYPE = PROC [arg: REF, structure: Structure ← NIL] RETURNS [Basics.Comparison];

BinaryCompareOp: TYPE = PROC [firstArg, secondArg: REF, structure: Structure ← NIL] RETURNS [Basics.Comparison];

StructuredToGroundOp: TYPE = PROC [structuredElt: REF, structure: Structure ← NIL] RETURNS [groundElement: REF];

For things like determinant and coefficient extraction

ElementRankOp: TYPE = PROC [arg: REF, structure: Structure ← NIL] RETURNS [CARDINAL];

For things like degree functions in euclidean domains, rank of matrices

Miscellaneous Operations

FlavorToRope: PROC [flavor: ClassFlavor] RETURNS [rope: Rope.ROPE];

GetProperty: PROC [structure: Structure, prop: ATOM] RETURNS [val: REF];

Retrieve from property list

END.