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(Last Mod: 27 November 2010 21:38:01 )
VERY, VERY INCOMPLETE
Recall in the discussion of basic modular arithmetic that when we devise a mathematical system we do not have to use the normal integers that we are already accustomed to from everyday life. We generally do so whenever we can because it makes things more familiar and convenient, but we are free to pick arbitrary symbols and then define any operations between them however we choose.
For instance, we can define a system of four symbols consisting of a circle, a square, a rectangle, and a diamond. We can then define an operation called "Fred" such that a circle Fredded with a rectangle yields a diamond while a square Fredded with a diamond yields a circle. We can define the "Fred" operation by simply enumerating every possible combination of inputs and the resulting output in a table. As strange as this may seem, this is actually a familiar concept to most people since, at least traditionally, children are taught addition using an "Addition Table" and multiplication using a "Multiplication Table". These tables are simply a way of defining these two operations in terms of all of the combinations of inputs (generally a restricted set of inputs, such as two single digit numbers) and the resulting output.
Our "Fred" may or may not have any practical uses; this depends on what properties it has and whether those properties are of any value to us. For instance, given the partial enumeration of the "Fred" operation given above, what results when a rectangle is Fredded with a circle? If you are like most people, you probably said that it is a diamond. The correct answer is, "I have no idea because it hasn't been defined." In concluding that it yields a diamond, you have assumed that the "Fred" operation acting on this particular set of symbols possesses a particular property, namely that it is commutative. But this is not a reasonable assumption and it was never stated that this property holds. Consider the division operation working on the set of real numbers - just because 6 divided by 3 is 2 does not mean that 3 divided by 6 is also 2. The reason is that division is not commutative.
So what are the properties that are of interest to us with regards to our symbol sets and operations that act on them? In general, we are interesting in such things as closure, identity elements, inverse elements, associativity, commutativity, and distributability.
A "set" is just a collection of symbols. Nothing more. In the example above, we defined a set that had four symbols, namely a circle, square, rectangle, and a diamond. Other sets might be the set of all integers (which is an infinite set because it has an infinite number of elements) or the set of all non-negative integers less than p (which is a finite set having exactly p elements). The elements of a set do not have to be numbers and they do not have to have any particular relationship between them.
An "operator" is a rule or set of rules that defines a relationship between all of the members of a set one pair at a time. In general, we can express an operation with the notation c = O(a,b) which means that if we take elements a and b from the set (these are said to be the "operands") and perform the operation indicated by O(), that the result is c. It is not required that c be an element of the set or that it even be defined. For instance, if the set is all integers and the operation is division of the first number by the second number, the O(6,3) is in the set, O(3,6) is defined but not in the set, and O(3,0) is undefined.
Another way of expressing an operator is to pick a specific symbol for it and place it between the two operands. For instance, we might define the # operator and express it as c = a # b.
In defining operations on sets, we frequently define the "addition" and "multiplication" operations, but it is important to remember that, in general, these are not the same addition and multiplication operations that we are familiar with from normal, everyday arithmetic. In general, they are arbitrary operations that we explicitly define and just happen to given the names "addition" and "multiplication" to. Never-the-less, there is a reason for doing so, namely that we define these operations so that, even though they are different that the operations we are familiar with, they still possess most of the same properties that we are familiar with. This is solely for our convenience in that it generally makes it easier for us to work with these operators working on various sets because most (but not necessarily all) of the familiar rules apply.
An algebra is nothing more than a Set combined with one or more Operators. This establishes the various manipulations that can be performed and what properties hold.
When we combine one (or more) operations with a set to form an "algebra," we establish certain properties that the algebra either does or does not have. For instance, a particular algebra may have an identity element but not have an inverse for every element in the set. If two algebras possesses the same collection of properties, then manipulations that can be performed in one algebra can be performed in the other. It is therefore useful to classify algebras according to which properties they do (or perhaps do not) possess. But before we can do that, we have to understand at least some of the various properties themselves.
The Closure property of an operation acting on a set of symbols is met if and only if, for every possible pair of operands from the set, the result is also a member of the set. The set of all integers is closed under multiplication because the result of multiplying any two integers yields another integer. In contrast, the set of integers is not closed under division because the result of dividing one integer by another may produce a result that is not an integer; for example, 3 divided by 2 yields a result that is not within the set of integers.
A more formal way of stating the closure property is that the set is closed under the # operation if and only if
c = a # b
yields a result c that is a member of the set for every possible pair of operands a and b chosen from the set.
This means that the O(a,b) always yields the same result as O(b,a). Using the alternate notation, a set is commutative under the # operation if and only if
(a # b) = (b # a)
every possible pair of a and b chosen from the set.
This means that the O(a,O(b,c)) always yields the same result as O(O(a,b),c). Using the alternate notation, a set is associative under the # operation if and only if
(a # b) # c = a # (b # c)
every possible selection of operands a, b, and c chosen from the set.
The identity element for an operation acting on a set of symbols is that member of the set (if it exists) such that the operation of any element and the identity element yields the original element. It is not required that all operators that can act on a set have the same identity element. For instance, in everyday arithmetic, the identity element under addition is 0 while the identity element under multiplication is 1.
In the generic case, the element 'I' is the identity element under a particular operation if and only if
a = I # a
a = a # I
for all possible selection of a chosen from the set.
Note that both forms are needed since we aren't making any claim that the algebra is commutative.
An inverse element, if it exists, is that element of a set such that the result of the operation between an element and its inverse element is the identity element for that operation. In other words, a-1 is the inverse of a under the # operation if and only if
I = a # (a-1)
I = (a-1) # a
The algebra satisfies the Inverse Law for a given operation if and only if every element in the set has an inverse that is also within the set for that operation.
Note that both forms are needed since we aren't making any claim that the algebra is commutative.
A subtle point here is that the notation a-1 does not mean "a raised to the power of -1." It is simply a notation meaning "the inverse element of a under the # operation."
The distributive property involves a set and two operators, which we will denote by # and $. An algebra satisfies the distributive property of # over $ if and only if
a # (b $ c) = (a # b) $ (a # c)
(b $ c) # a = (b # a) $ (c # a)
Note that both forms are needed since we aren't making any claim that the algebra is commutative with respect to the # operator.
As mentioned previously, if two algebras possesses the same collection of properties, then manipulations that can be performed in one algebra can be performed in the other. There are a few collections of properties that sufficiently common and/or useful that they have been given specific names. The names chosen are quite arbitrary, so don't attempt to read anything into the name based on the everyday meaning of the word used.
This is an algebra involving a set and one operator. The algebra is a Group if and only if, under that operator, it:
An Abelian Group (also known as a Commutative Group) is a Group that is also commutative under the operator. Hence an algebra is an Abelian Group if and only if, under that operator, it
This is an algebra involving a set and two operators. An algebra is a Ring if it is an Abelian Group under the first operator, and satisfies the properties of closure and associativity under the second operator, and for which the second operator is distributive over the first.
This algebra also involves a set and two operators. An algebra is a Field if it is an Abelian Group under the first operator, if the set consisting of all the elements of the complete set less the identity element of the first operator is an Abelian Group under the second operator, and for which the second operator is distributive over the first.
http://www.csee.umbc.edu/help/theory/group_def.shtml (last accessed 26 Jul 07).
http://en.wikipedia.org/wiki/Algebraic_structure (last accessed 26 Jul 07).