# Questions tagged [associativity]

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

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### Is every associative $n$-ary operation with an identity element induced by a monoid?

Given any $n$-ary operation $*$ on a set $X$, an identity element for $*$ is an element $e \in X$ such that $x*e*e*...*e=e*x*e*e*...*e=...=e*e*...*e*x=x$ ($n-1$ $e$s in each product) for all $x \in X$....
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### Associativity of operation of finite set

I'm studying Basic Algebra 1 by nathan jacobson at home. I found that (i) implies for every element in G there exist right inverse and left inverse. I think I should show is associativity of ...
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### Has the associative property been generalized to k-ary functions?

I've been exploring why the associative property is so interesting to mathematicians. Along the way, I have found the rather obvious fact that it only works on binary operations. It needs a concept ...
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### What's the difference between $-81^{3/2}$ & $(-81)^{3/2}$?

Calculating $81^{3/2}$, I got $729$ (not saying it is correct, but I am trying :) ). Would $-81^{3/2}$ just be the opposite ($-729$) and does it make a difference if $-81$ was placed inside a pair of ...
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### Why is the associative property so special to mathematicians?

A few years back I came across an article on quantum physics in Quanta Magazine. It described the work of Cohl Furey trying to plumb the secrets of the universe using octonions. The article explains:...
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### How to prove that convolution of sequences is associative?

Let {$a_n$} and {$b_n$} be finite real sequences with $n\ge0$. Convolution ($\ast$) of two sequences defined as $$\{a_n\}\ast\{b_n\}=\{\sum_{i=0}^{n} a_ib_{n-i}\}.$$ The convolution of three ...
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### Fewest applications of associativity

By repeatedly applying the basic associativity law $(x+y)+z = x+(y+z)$, one can get from any one expression with binary addition to any other with the variables in the same order. Specifically, given ...
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### Investigating and Generalizing Octonionic Nonassociativity

A possible multiplication convention for the octonions is given by the following 7 sets of integers from 1 to 7. {1,2,3},{1,4,5},{1,6,7},{2,4,6},{2,5,7},{3,4,7},{3,5,6}. Notice that each of the seven ...
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### Associativity and piecewiseness

I have this confession: Piecewise definitions have always considerably annoyed me (with certain exceptions). For $x,y\in (-1,1),$ let $x\circ y := xy -\sqrt{(1-x^2)(1-y^2)}\in(-1,1).$ (The part that ...
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### Why the universal enveloping algebra of a Lie algebra is an associative algebra with a unit?

I have seen the definition of the universal enveloping algebra of a Lie algebra is an associative algebra with a unit which satisfies a universal property (always exists and is unique up to ...
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### How to show nonassociativity of the positive rationals under a binary operation defined in terms of max and min?

Consider $\mathbb{Q}^+$ with the usual $\leq$ relation and the binary operation $\circ$ defined as: $$p \circ q = max(p,q) + \frac{1}{2} min(p,q)$$ A book that I'm reading states that the operation \$\...