Well defined binary operation
In this context, we discuss some invariants that can be associated with the identity. For full proof, refer: Associative implies generalized associative. When a binary operation is associative, it turns out that we can drop parenthesization from products of many elements. That is, given an expression of the form:. The result is proved by induction, with the base case following from the definition of associativity. For this reason, we always use infix operator symbols for associative binary operations, and often even drop the operator symbol, so that the expression is just written as: Also, the re-parenthesization identities i.
The associativity pentagon is a pentagon whose vertices are the five different ways of associating a product of length four, with an edge between two vertices if moving from one to the other requires a single application of the associative law. This is a cyclic pentagon. The associativity pentagon is significant because, loosely, it generates all relations between the different ways of applying the associativity law to re-parenthesize expressions.
It also helps to prove results about the set of left-associative, middle-associative, and right-associative elements. It is also related to the associator identity. In the presence of associativity, it is possible to unambiguously define positive powers of any element. Explicitly, is the -fold product. The powers satisfy the usual laws of powers: Note that this also implies that all powers of commute with each other.
Note that to define powers, we do not actually need global associativity, but only power-associativity: Suppose is a commutative unital ring and is a possibly associative, possibly non-associative algebra over. In other words, is a -module and is a possibly associative, possibly non-associative binary operation. Note first that in order to verify the associativity of multiplication globally, it suffices to verify associativity on a generating set for the additive group of as a -module.
This is because the associator function is -linear in each input:. In the special case that is freely generated as a -module, the following test works for associativity: If form a freely generating set of as a -module, and denote the structure constants , then associativity is the following identity for all:.
For a non-associative ring with multiplication , we can define the associator as:. An element is said to be left-associative or left nuclear with respect to a binary operation if any ordered triple starting with that element associates. The set of left-associative elements in any magma is a subsemigroup called the left nucleus. If the magma contains a neutral element, it is a submonoid. Left-associative elements of magma form submagma. An element is said to be middle-associative or middle nuclear with respect to a binary operation if any ordered triple with that element in the middle, associates.
The set of middle-associative elements in any magma is a subsemigroup called the middle nucleus. Middle-associative elements of magma form submagma.
An element is said to be right-associative with respect to a binary operation if any ordered triple ending with that element associates. The set of right associative elements in any magma is a subsemigroup called the right nucleus. Right-associative elements of magma form submagma. An element is said to be associative if it is left, middle and right associative. Now let's apply this! If I give you two numbers and a well defined operations, you should be able to tell me exactly what the result is.
If you tell me the answer is 5, I could just say, "Nope, the answer is We don't mean multiplication , although we certainly can use it for that. But normally, we just mean "some operation". When we do mean multiplication we say so. Now that we understand sets and operators, you know the basic building blocks that make up groups. But it is a bit more complicated than that.
We can't say much if we just know there is a set and an operator. What more could we describe? We need more information about the set and the operator. This is why groups have restrictions placed on them. That is, they have more properties. The group contains an identity. If we use the operation on any element and the identity, we will get that element back. For the integers and addition , the identity is "0". The symbol for the identity element is e , or sometimes 0.
But you need to start seeing 0 as a symbol rather than a number. It's defined that way. In fact, many times mathematicians prefer to use 0 rather than e because it is much more natural. The group contains inverses. If we have an element of the group, there's another element of the group such that when we use the operator on both of them, we get e , the identity.
In just the same way, for negative integers, the inverses are positives. In fact, if a is the inverse of b, then it must be that b is the inverse of a. Make a note that while there exists only one identity for every single element in the group, each element in the group has a different inverse.
The notation that we use for inverses is a You should have learned about associative way back in basic algebra. All it means is that the order in which we do operations doesn't matter. Notice that we still went a All that changes was the parentheses.
We'll get back to this later Closed under the operation. Imagine you are closed inside a huge box. When you are on the inside, you can't get to the outside. In that same way, once you have two elements inside the group, no matter what the elements are, using the operation on them will not get you outside the group.
This is what we mean by closed. It's called closed because from inside the group, we can't get outside of it. And as with the earlier properties, the same is true with the integers and addition. So, if you have a set and an operation , and you can satisfy every one of those conditions, then you have a Group.
Way back near the top, I showed you the four different operators that we use with the numbers we are used to:. When we subtract numbers, we say "a minus b" because it's short.
But what we really mean is "a plus the additive inverse of b". The minus sign really just means add the additive inverse. But it is crazy saying that over and over again, so we just say "minus". Can you take a guess at what division is?
In the same way, it just means "multiply by the multiplicative inverse". Well this is an odd example. But let's try out the three steps. Let's find the identity element.
Well, that shouldn't be too hard. If we add 0 to anything else in the group, we hope to get 0. Now we need to find inverses. Well, again, we only have one element. So what's the inverse of 0? Since we've tried all the elements, all one of them, we're done. Finally, is it closed? You bet it is. Back to the four steps. First, is there an identity? Well, this is going to be easy, there are only three possibilities. So let's start off with 1. Since we have found an inverse for every element, we know the group is closed with respect to inverses.
Well, since we have only 2 numbers, we can try every possibility.