>My math background made me realize that each object could have several algebras associated with it, and there could be families of these algebras
Could someone explain what he meant here?
Could someone explain what he meant here?
The algebra of arithmetic is the most familiar, and it lets us determine that f(x) = 1 + (1 + (1 + x)) is equivalent to g(x) = ((1 + 1) + 1) + x for any integer x. In this case, the relevant algebraic law is associativity.
One abstract way to think about integers is as sets of equivalent arithmetic trees, so "3" is a stand in for the set {(1 + 1) + 1, 1 + (1 + 1)}.
Other kinds of objects that can appear in a program can have similar kinds of equivalences under various operations.
As a less familiar example, take axis aligned bounding boxes. Translation and scaling both map an axis aligned bounding box to another axis aligned bounding box. Intersection maps two bounding boxes to one smaller bounding box. "Convex closure" maps two bounding boxes to a larger bounding box--specifically, the smallest bounding box that encloses both of them.
There are then various algebraic laws that let us determine that differently expressed combinations of these operations are in fact equivalent.
If "∧" represents intersection, and A, B, and C are bounding boxes, then f(A, B, C) = (A ∧ B) ∧ C and g(A, B, C) = A ∧ (B ∧ C) are equivalent because intersection is associative.
If we apply the same translation to two bounding boxes and then form the intersection of the results, this is equivalent to intersecting the boxes, and then translating the results.
There are also interesting equivalences that relate intersection and closure.
This kind of broadly construed "algebraic" manipulation can be used to rewrite one program into an equivalent program that executes more efficiently, and this is one way of thinking about what an optimizing compiler is doing when it optimizes a program.
The auto "fits" into each of these different problem domains.
That probay doesn't clarify the connection, but at least you'll be pointed in the right direction! :)
"High school algebra" is usually limited to the study of real numbers and the operations of addition and multiplication.
In upper division undergraduate mathematics, you learn about other kind of algebras. For example, there is the set of regular polygons with operations such as rotation and reflection.
In OO, a class of objects defines the operations (methods) that are valid, and thus, a class is an algebra, and a set of classes is a family of algebras. I suppose that you would need a form of multiple-inheritance in order to have a family of algebras for an object.