Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [associativity]

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$. This is a key property of groups, rings and fields.

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Does commutativity imply Associativity?

Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples? NOTE: I wasn't sure how to tag this so feel free…
Eugene
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Does associativity imply commutativity?

I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation,…
Aritra Das
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Is there an intuitive reason for a certain operation to be associative?

When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation $$x*y=\frac{xy}{x+y+1}$$ (defined on the positive real numbers) is associative. At first I considered this to be false, because the expression is so…
Zirui Wang
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Is there an easy way to see associativity or non-associativity from an operation's table?

Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a & e & d & b & a & c\\ b & d & c & e & b & a\\ c & b & e & a &…
celtschk
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Real life examples of commutative but non-associative operations

I've been trying to find ways to explain to people why associativity is important. Subtraction is a good example of something that isn't associative, but it is not commutative. So the best I could come up with is paper-rock-scissors; the operation…
Brian Rushton
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Non-associative commutative binary operation

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative binary operation I have in mind is the…
Rodrigo
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Why worry about commutativity but not associativity in The Fundamental Theorem of Arithmetic?

A common statement of The Fundamental Theorem of Arithmetic goes: Every integer greater than $1$ can be expressed as a product of powers of distinct prime numbers uniquely up to a reordering of the factors. Now the statement makes a point of…
Mike Pierce
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How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a counterexample would be to let $x_1*x_2*\cdots*x_n$ to…
hmakholm left over Monica
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Expressing associativity with only two variables

I'm wondering if it is possible to axiomatize associativity using a set of equations in only two variables. Suppose we have a signature consisting of one binary operation $\cdot$. Is it possible to find a set $\Sigma$ of equations containing only…
Tom
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How to prove that the cross product doesn't satisfy any kind of generalized associativity?

It's well known that the cross product in $\mathbb{R}^3$ doesn't obey the associative law of $$ A \times (B \times C) = (A \times B) \times C $$ We can define a "Generalized Associative Law" as an expression involving equation two sets of $N$ same…
Sidharth Ghoshal
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Are all algebraic commutative operations always associative?

I know that there are many algebraic associative operations which are commutative and which are not commutative. for example multiplications of matrices as associative operation is not commutative. I need to know about inverse of this! I mean is…
IremadzeArchil19910311
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Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "brackets actually exist" and "associativity is really…
Shaun
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Are all associative operations essentially function composition?

It is well known that function composition is associative. Is the converse true? Is any associative operation essentially expressible as function composition? If not, what's an interesting example of an associative operation that is not expressible…
xyz
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Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) composition is associative: $$h \circ (g \circ f) = (h…
Val
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Superassociative operation

Background: Addition and multiplication are associative, but exponentiation is not. Question: Does an operation $\circ_1:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ exist such that $$\circ_i(x,y)=\underset{y\text{…
Carucel
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