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

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Can Euler's formula be extended to Octonions and higher Cayley-Dickson algebras that are non-associative?

Although question is pretty self explanatory, I know that $e^{i\theta}=\cos\theta+i\sin\theta$ and $i$ can be replaced by a unit pure quaternion (which is a root of $-1$ like $i$). Can we do the same ...
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How to define a taxonomy of non associative operations?

Let $A$ be a set, and let $a,b,c\in A$. Let also $\circ: A\times A\rightarrow A$ be a binary operation on $A$. We agree as usual to write $a\circ b$ to mean $\circ(a,b)$. We say that $\circ$ is ...
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+100

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
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30 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
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54 views

If $x * x = e$ for all $x$ and $(x_i * x_j) * (x_j*x_k) = x_i*x_k$, then $*$ is associative

This should be simple, but for some reason I get stuck on this. Let $G = \{x_1, \ldots x_n\}$ be a set equipped with operation $*$ satisfying the following : 1) $G$ has an identity element $e$ with ...
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1answer
51 views

Which is the property of the functions that correspond to this definition/examples?

I'm looking for a definition for a particular function(-input) property. Considering a function $f$ that takes as input a list of elements and produces in output just one element, which is the ...
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34 views

How to prove that: If two binary operations are anti-isomorphic and one of them is associative then the second one also will be associative?

We know what is called an anti-isomorphic operation on a set S. it is just a one two one $ g $ function mapping from $S$ to $S$. $ g: S \rightarrow S$. and it satisfy this condition $ g(xy)= ...
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Are there associative magic squares of any size except $4k+2$?

An associative magic square is a magic square with the additional property that numbers symmetric to the center sum up to $n^2+1$. For example, the square ...
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87 views

Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...
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56 views

Associative Lie algebra without Jacobi identity

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold? 2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements? 3) ...
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40 views

Is the number of associative $n$-ary algebraic operations on a finite set with 2 cardinality always 8?

We know that if $n = 2$ then the operation is called a binary operation. $ \circ $ on set $X$ is a function $\circ : X \times X \rightarrow X$. And the number of all associative binary operation on a ...
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Example of operation that is tree associative, but not generally associative

In a lot of algorithms using trees, we need the property that when folding $2^n$ elements with some operator $+$, we can do the first half of $2^{n-1}$ elements and the second half independently. Eg ...
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29 views

Associative property for series

Are those equation always valid: $$\sum a_n + b_n = \sum a_n + \sum b_n$$ $$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}$$
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Name of an aglebraic structures $(A,*,\cdot)$ weaker than semirings.

I have a set $A$ with two binary operations on it $(A,*,\cdot)$ STRUCTURE A $(A,*)$ is not commutative, is not associative, it has not an identity $(A,\cdot)$ is a commutative group $(a*b)\cdot ...
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Proving $K$ is a group

Now I have proved certain things are a group before, and I know that it requires: 1)Associativity 2)Inverse 3)Identity But here I have such a strange thing that I wanted to clarify that I am doing ...
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78 views

Show that * is associative

Could you show me how to prove the following to be associative? Please take me through the process step by step. $$a*b=a+b+2ab$$ Where $*$ is a binary operation and $a$ and $b$ are real numbers. I ...
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1answer
44 views

Proof of a corollary about associativity of (differential) convolution operater

I m working on the proof the following corollary for ages... I would appreciate any help!! Cor: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define the convolution operator $*$ for the ...
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1answer
21 views

Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as: $\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ...
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1answer
35 views

Is the operation associative

Is it known that the multiplication of matrices is a associative operation ? So,is the relation $(A \cdot B) \cdot C=A \cdot (B \cdot C)$ true?? ($A,B,C$ are matrices)
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Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I'm not sure ...
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38 views

The condition for associative property

Thank in advance. Is there the condition for associative property in closed number system. It comes from a question. A number system is closed, associative, commutative for some operation, then the ...
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158 views

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 ...
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1answer
26 views

Algebra of pseudo-differential operators

The class of pseudod-ifferential operators form an associative algebra of Fourier integral operators. Moreover, given symbols $a,b,c\in C^\infty$ (each associated to some pseudo differential ...
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tax and discount problem

Problem : In the total purchase amount $z, x\%$ is tax and $y\%$ is discount. Even if the tax is applied first and then discount or if discount is applied and then tax, the final amount is always ...
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Proof of The Associative Law and The Commutative Law.

The associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...
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30 views

Can we have an associative form of “octonions” and hypercomplexes, if we eliminate division?

I'm interested in hypercomplexes, or number systems with many square roots of $-1$. Now, I know that quaternions are non-commutative, but associative. I'm wondering if it's possible to have a number ...
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1answer
41 views

Why are exponents not associative?

I ran into something that seemed odd to me today: exponents are not associative. The following equation sums that up: $$ 10 * 2^{5x} \not\equiv 20^{5x} $$ Why is this the case? Is there some ...
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107 views

Does associativity justify $(f^{-1}gf)(f^{-1}hf) = f^{-1}gff^{-1}hf$?

I'm self-studying abstract algabra (Herstein) and while working on an easy problem became uneasy with a step in my derivation. Given the symmetric group $S_n$ whose elements are bijections $f: S \to ...
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1answer
37 views

Splitting up bracket terms

I found a statement saying: Let $\circledast $ be an associative binary operation on a set $\mathbb{X}$. A bracket term of length n, consisting of n elements $a_1, ..., a_n$ and arbitrary brackets, ...
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Is convolution associative with regards to the complex unity?

Setup: I need to do a convolution with the function $\cfrac{i}{x}$, and I would like to get rid of the $i$. My functions to be convolved are all real valued. According to the ever-failable wikipedia, ...
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115 views

How to correctly do a division using a slash?

What's the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$? Basically, if I place a slash, will I divide by what's directly behind the slash ...
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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 ...