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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Proof verification for showing that the sum of real numbers is independent of parenthesis using induction

Let $S_n = x_1+x_2+\ldots + x_n$ be the sum of n real numbers and $P(n)$ be the property that $S_n$ is independent of parenthesis and E$:= \{ i \in \Bbb N : P(i) \}$ , then, $$ (S_1 = x_1 = (x_1) ) \...
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How many associative binary operations on the integers does $+$ distribute over?

I am interested in binary operations $\mid: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ which satisfy: Associativity: $a \mid (b \mid c) = (a \mid b) \mid c$ $+$ distributes over $\mid$: $(a \mid b) ...
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Prove that $a + (b+(c+d)) = (a+b) + (c+d)$

I'm working through Spivak's Calculus and the first section is covering associativity of addition. I am only given $P1: a + (b + c) = (a + b) + c$. I obviously know this intuitively but I cannot make ...
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1 answer
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Show that if $f$ is a homomorphism then the set of invertible elements $M^\times$ is commutative

I have to show the following. Let $M$ be a monoid. If $M \to M$, $f: a \mapsto a^2$ is a homomorphism, then $M^\times$ is commutative. So, if $f$ is a homomorphism, then for all $a,b \in M^\times$ we ...
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Field theory and distributive law.

How do any subset of a Field inherit the properties-commutativity,associativity and specially Distributive law over the same operationsas that of field?Is there any intuitive way to understand that ...
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How can I prove the associative property of this set?

Let $(G,+)$ be a group. Let $a$ belong to $G$. Let $C(a)=\{ g \in G \mid g+a=a+g\}$. Does $C(a)$ satisfy the associative property? WARNING (1) : It is not valid to prove that $C(a)$ is a subgroup. ...
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What constraints does associativity put on a real function $f(x, y)$?

We are familiar with the algebraic meaning of associativity. But suppose we have a real function $f(x, y)$. What constraints does $$ f(f(x,y),z) = f(x,f(y,z)) \label{1}\tag{1} $$ put on $f$? Can they ...
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Is it necessary to check every possible triple of values to confirm a binary operation is associative?

Is it possible to use less than n^3 (where n is the cardinality of the set the operation works on)? That is, is there some way ...
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6 votes
5 answers
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What is the mistake with this proof that commutativity implies associativity?

I tried to show that commutativity of addition implies associativity. For this I assumed that there is no associative property and $ a + b + c $ should be interpreted as $(a + b) + c $ . $$(a + b) + ...
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Prove the associativity of $(\Bbb Z_7,\oplus)$. [closed]

I am reading this Problem 14.4.1. We want to prove that $(Z_7,\oplus)$ is group. I have difficulty proving associativity axiom. The solution reads Associativity: Let $a\in\mathbb Z_7,$ $b\in\mathbb ...
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Is $\gcd(\gcd(a,b),\gcd(c,d)) = \gcd(\gcd(a,c),\gcd(b,d))$ another instance of GCD Associativity? [duplicate]

The question is as is in the title: Is $\gcd\bigg(\gcd(a,b),\gcd(c,d)\bigg) = \gcd\bigg(\gcd(a,c),\gcd(b,d)\bigg)$ another instance of GCD Associativity? I know that $$\gcd\bigg(e,\gcd(f,g)\bigg)=\...
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Is Generalized version of associative law in Halmos's set theory superfluous? How can we write a generalized commutative law?

In Halmos's naive set theory, section 9 (families), page 35 there is a part about generalized version of the associative law. It reads as follows: The algebraic laws satisfied by the operation of ...
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Associative Law - total number of combinations satisfying the associative law [duplicate]

I'm stuck in a problem and need help. Associative law states that (a + b) + c = a + (b + c) If there are 4 elements, there are 5 arrangements satisfying the property. (a + b) + (c + d) ((a + b) + c) + ...
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If the associative law doesn't hold, can we define $a^n$? [duplicate]

I am reading "Higher Algebra" by A. Kurosh. The following sentence is in this book: Analogously, the associative law of addition leads to the concept of a multiple, $na$, of the element $a$ ...
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Associahedron, but with swaps

The associahedron has edges of the form $a(bc)\rightarrow (ab)c.$ But I also want to include the possibility of swapping adjacent entries by doing operations like $a(bc) \rightarrow a(cb).$ I was ...
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Derivations of soluble finite-dimensional associative $K$-algebras

Let $K$ be a field and $A$ a finite-dimensional associative unitary solvable $K$-algebra such that a self-centralizing radical complement $T$ exists which possesses a basis $\{e_1,...,e_n\}$ ...
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How do I check that this composition is associative?

I have the following problem: Let $R$ be a ring, and $M$ a monoid. we have the multiplication on $R[M]$ given by: $$\left(\sum_{m\in M} a_m\cdot m\right) \cdot \left(\sum_{m\in M} b_m\cdot m\right)=\...
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Cayley tables and associativity [duplicate]

I don't see how can one check associative property in a Cayley table. For contrast, one can find identity element, inverses and commutativity. But how you check associativity? I understand that ...
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Commutator relations in associated algebra.

Consider the free associated algebra over some field $k$ $$k\langle x_1,\cdots,x_n\rangle.$$ Order the generators $x_1<\cdots<x_m$. I can define a complete bracket of an ordered monomial $x_{i_1}...
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Proof of Matrix Cross-Multiplication Distributive Property

I am trying to prove that $A\times(B+C)=A \times B + A \times C.$ I have managed to prove this by finding the LHS $$\begin{pmatrix} a_{1}\\a_{2}\\a_{3} \end{pmatrix} × \begin{pmatrix} b_{1}+c_{1} \\...
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3 answers
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Is it necessary to show 0 is included in a set to show it is a vector subspace?

To show a set is a vector subspace, I see it´s necessary to prove a) An addition property: if $x$ and $m$ are both elements of the set, then $x+m$ must also be an element of the set for it to be a ...
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In Polish/Prefix notation of binary operations, are all expression unambiguous w.r.t. which operator is applied first?

Take binary operator ~ and a chain like a ~ b ~ c. It is ambiguous as it can mean either of: ...
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How to read/verbalize lambda expressions? What is `(λx.λy).M`?

In lambda calculus, application is left associative. That is, if M, N, and P are expressions,...
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2 votes
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Is quaternion multiplication associative with matrix multiplication?

Denoting a quaternion as $\mathsf{q}= (q_0,\mathbf{q})$, the quaternion product is given by: $$ \mathsf{q} \otimes \mathsf{p} = \begin{pmatrix} q_0 p_0 - \mathbf{q}\cdot\mathbf{p} \\ q_0 \mathbf{p} + ...
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1 answer
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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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4 votes
2 answers
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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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5 votes
1 answer
248 views

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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3 votes
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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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-1 votes
1 answer
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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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4 votes
1 answer
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What does "associative up to homotopy" mean for $A_\infty$-algebras?

I'm reading Keller's Introduction to A-infinity algebras and modules to learn about $A_\infty$-algebras. For reference, an $A_\infty$-algebra $A$ is a graded $k$-vector space $A = \bigoplus_{i\in\...
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Create smaller products while guaranteeing uniqueness - Godel encoding

I was reading about Godel Numbering I was wondering if it is possible to create a unique number in a way (i.e. as a product), but with that the requirement the number does not grow at the rate of the ...
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1 vote
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Number of steps needed to show that a binary operation is associative on n operand

I was trying to prove a binary operation is associative on a given number of the operand. I did it, but then I checked the textbook. I took 5 lines, but the book proved it in 6 lines using a precise ...
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2 votes
1 answer
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Associativity of binary operations on a two-element underlying set (is there a pattern?)

The overall problem is to establish, which binary operations on a two-element set $A=\{a, b\}$ are commutative and associative. There are 16 of them altogether, obviously, analogous to operations on ...
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Quickly checking associativity of a possible group action

In Artin, Algebra, the author says that it is usually clear in a given example of a group action that the axioms for it hold. The simplest example given is the action of the group $\{1,r\}$ on $\...
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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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1 vote
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Galois Field $\mathrm{GF}(4)$

I am trying to prove associativity for the following. Addition: $$ \begin{array}{c|cccc} + & 0& 1& B & D \\ \hline 0& 0 & 1 & B & D \\ 1 & 1 & ...
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Proving Associativity of Matrix Multiplication.

Although this question is already asked and also answered which I am thinking is a bit algebraic..(simply put tough for me ;-) ) hence I had started thinking of in direction of proving it using ...
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2 votes
0 answers
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Derivation of Least Squares / Associativity of vector multiplication

I tried to solve Linear Least Squares and I am kind of stuck / confused about the property of vector multiplication/associativity/commutativity. The way I defined my problem is the following: Let $\...
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0 votes
1 answer
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Understanding on Artin's proof of the generalised associative law for associative binary operation

On the 2nd edition of Artin's Algebra, the writer uses proposition 2.1.4 to imply that generalised associative law works for associative binary operation: Proposition 2.1.4 Let an associative law of ...
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Internal binary operation

Dirichlet Convolution. If $f,g:\mathbb {N} \to \mathbb {C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution $f ∗ g$ is a new arithmetic ...
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6 votes
1 answer
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Are inverse unique in unital algebra?

Let $A$ be a unital algebra over a field $F$ with unity $1$. If $a,b,c\in A$ such that $ab=ba=1=ac=ca$, does this imply that $b=c$? We have $c(ab)=c$. However, algebras are not necessarily associative ...
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Inverses without associativity

This question might be a little too open-ended, so I will not be chagrined if it is closed and will appreciate suggestions to make it more specific in the comments. A little background: I recently ...
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2 answers
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How to show the associativity in this group?

The group law is given by $a \oplus b = \frac{a+b}{1+a\,b}$ for $a,b \in G = (-1,1)$ For associativity I guess I have to show $(a \oplus b) \oplus c = a \oplus (b \oplus c)$, hence: $$\begin{equation}\...
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1 vote
1 answer
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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 $\...
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2 votes
2 answers
150 views

Are all morphisms in category theory are actually some kind of mappings? What is the importance of associativity rule in category theory?

I always found the identity and the composition rule trivial while thinking about categories. But I think the associativity rule is NOT trivial and it is hard to create a composition table that obeys ...
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3 votes
3 answers
265 views

Showing that the Klein-four group is associative without checking 64 options

Showing that the Klein-four group is associative without checking 64 options I don't know how to go about this. I could use the fact it's abelian to eliminate a few duplicates, but many options ...
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