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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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priority of operations in function composition is backwards

I feel like the priority of function composition is backwards, and I would like to have a deep understanding of the phenomenon. I do understand that function composition reads right to left: $$(f\circ ...
Victor Daniel's user avatar
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1 answer
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Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]

Associativity is: $$(a * b) * c = a * (b * c)$$ Alternativity is: $$a * (a * b) = (a * a) * b$$ $$(a * b) * b = a * (b * b)$$ Bol loop is: $${\displaystyle a(b(ac))=(a(ba))c}$$ $${\displaystyle ((ca)b)...
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Proof for associativity of a modified "matrix multiplication"

I'm trying to prove the associativity of a modified form of matrix multiplication (which is defined below) and I found the following proof which I'm confused about: For matrices $W_i, W_j, W_p$, to ...
Hugh Mann's user avatar
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Why is 1+2+3 = (1+2)+3 [duplicate]

Not sure if this is a stupid question, apologize if it is. I am curious why we can add the first 2 numbers, then add the third one when doing addition of 3 numbers. There is a similar (IMHO) question, ...
Steve Lau's user avatar
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A pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ such that $f$ and $g$ are idempotent, commute with each other and $f \times g$ is bijective

The question The question is: Does there exist a pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ satisfying the following properties? $f$ and $g$ are idempotent, meaning that $\forall n \in \...
Smiley1000's user avatar
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why is this associative?

I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already 😅 The fourth exercise asks me to determine whether the following operation is compliant with the associative ...
invalid syntax's user avatar
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1 answer
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$(ab)c + a(bc) = 2 b (ac) \implies^? x(yz) = (xy)z$? [closed]

Consider some unital commutative algebra $A$ such that for all its elements we have $$(ab)c + a(bc) = 2 b (ac) $$ Does this imply the algebra is associative ? or in symbols : $$(ab)c + a(bc) = 2 b (ac)...
mick's user avatar
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Proving Associativity of the Sum in a Space of Infinite Sequences with Non-Zero Initial Element

Consider the set $V$ consisting of all infinite sequences $a = (a_0, a_1, \ldots)$ where each $a_i \in \mathbb{R}$ and $a_0 \neq 0$. How can we demonstrate that the operation $(a + b) + c = a + (b + c)...
Fernand's user avatar
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Do we ever reason about a non-associative algebra without embedding it in an associative algebra?

This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math. ...
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Can any binary operator be turned, through an associative operator, into another associative operator?

Motivation: let $\circ:X^2\to X$ be some binary operator, and let $+:X^2\to X$ be some commutative operator. Then $$\star:X^2\to X:(x,y)\mapsto (x\circ y)+(y\circ x)$$ is commutative. I was wondering ...
Sam's user avatar
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When can a partial associative operation be extended?

Let $X$ be a set with a partial operation $\cdot$ which is associative in the sense that if $x, y, z \in X$ and $x \cdot y$ and $y \cdot z$ are both defined, then $(x \cdot y) \cdot z$ and $x \cdot (y ...
I Eat Groups's user avatar
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Showing matrix multiplication is associative via linear mappings.

Exercise. Prove that matrix multiplication is associative. In other words, suppose $A, B$, and $C$ are matrices whose sizes are such that $(AB)C$ makes sense. Explain why $A(BC)$ makes sense and prove ...
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Is this 3D algebra $T$ power-associative?

Before reading this question it is essential that you understand power associativity https://en.wikipedia.org/wiki/Power_associativity In particular a commutative algebra does not necc imply a power-...
mick's user avatar
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4 votes
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Abstract formulation of associativity

Say we are given a binary operation $f$ on a set $X$, that is, $$ f : X \times X \to X. $$ Denote by $\text{Id}$ the identity map on $X$. We say that $f$ is associative if, for all $x, y, z \in X$, we ...
markusas's user avatar
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How do you multiply algebraic expressions or matrices with more than 3 variables?

How do you multiply expressions that have more than 3 variables? For example, how would you multiply: $a \cdot b \cdot c \cdot d $ if associativity only defines the multiplication of 3 variables? ie....
Alice 's user avatar
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Does the associative property for multiplication hold when there are more than 3 factors multiplied together? [duplicate]

The associative property for multiplication states that: $(A \times B) \times C = A \times (B \times C)$ What happens in the case where there is an expression with more than $3$ factors? For example, $...
Kuskuba's user avatar
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Associativity on concatenation and removal

Let * be an operation on two ordered lists of numbers, where each list has no duplicates. The operation concatenates them and then removes any adjacent identical values recursively. I want to know if ...
Adam's user avatar
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Asociativity of disjunction in elementary toposes

This is a continuation from a previous question of mine about Logical disjunction in elementary toposes I managed to prove conmutativity of both $\curlywedge$ and $\curlyvee$ (conjunction and ...
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16 votes
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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" ...
Sidharth Ghoshal's user avatar
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Is there a quick and economical way to guarantee matrices/vectors with this binary operation in place of addition are associative?

I have an algebraic structure that I'll define with the ordered triple $(\{0, 1 \}, \oplus, \times)$ such that the following properties hold: $$ 0 \oplus 0 = 0$$ $$ 1 \oplus 1 = 1 $$ $$0 \oplus 1 = 1 ...
Nate's user avatar
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Multiplication of more than 3 factors (Associativity)?

This is an elementary math question. How to interpret multiplication of positive integers geometrically when the number of factors exceed three? length * width (area of a rectangle) length * width * ...
estrella's user avatar
1 vote
1 answer
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Associative definition of ordered pairs?

There are a few definitions of ordered pairs $(a,b)$. Wiener's def: $(a,b)_{W} = \Big\{ \big\{ \{a\}, \emptyset \} \big\}, \big\{ \{b\} \big\} \Big\}$ Kuratowski's def: $(a,b)_{K} = \big\{ \{a\}, \{...
Mykola Hordeichyk's user avatar
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Proving that a group homomorphism preserves associativity

I felt this was trivial, but I wanted to make sure. The proofs I've read which show that the image of a group homomorphism is a subgroup of its codomain only prove that closure, identity, and inverse ...
Nate's user avatar
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Is this proof about associativity correct, and can we simplify it?

I want to prove that associativity $$ a \circ (b \circ c) = (a \circ b) \circ c $$ implies $P(a_1, \dots,a_n) = P'(a_1,\dots,a_n)$ for all $n \geq 1$ and all $a_1,\dots,a_n$. $P(a_1, \dots,a_n)$ and $...
God bless's user avatar
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3 votes
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Prove that Matrix Multiplication is Associative. [duplicate]

Q: In this exercise we show that matrix multiplication is associative. Suppose that $A$ is an $m*p$ matrix, $B$ is a $p*k$ matrix, and $C$ is a $k*n$ matrix. Show that $A(BC)=(AB)C$. My solution: . $...
Eric's user avatar
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Speculative- Associativity and Information Loss

In Axler's Linear Algebra Done Right, the following problem (1.B.6) was posed: Let $\infty$ and $-\infty$ denote two distinct objects, neither of which is in $\mathbb{R}$. Define an addition and ...
Then-Brief-864's user avatar
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Associativity of a semidirect product

I have the following problem. Let $$ 0\to A\to G\to Q\to 1 $$ be a central group extension with $A$ abelian. Assume that this extension splits, i.e., $G\cong A\rtimes Q$. Now consider an action of ...
Igor Sikora's user avatar
5 votes
1 answer
93 views

How to describe all semigroups $(S, \, \cdot)$ based on a choice operation?

Let $S$ be a non-empty set. We say that a binary operation $f \, \colon S \times S \to S$ is a choice operation if it always returns one of its arguments. In other words, $\forall \, a \in S \, \colon ...
John McClane's user avatar
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1 vote
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Associativity of tensor product using the universal property (and not elements)

My question I thought the author of this question was really pushing for what I'm asking here and I don't think it was ever fully addressed in other questions (happy to be wrong!). My main question is ...
cheyne's user avatar
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3 votes
1 answer
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The succintness of the definition of associativity

I'm trying to get into linear algebra, but more importantly, I'm trying to get into a topic of mathematics outside of school, and now my brain doesn't have to prioritize and filter questions for ...
MadEmperorYuri's user avatar
3 votes
0 answers
211 views

Does the percentage of associative operations on a finite set decrease monotonically towards zero?

In this answer, André Nicolas proves that it is rare for a binary operation on a finite set to be associative, in the following sense: if $A_n$ denotes the number of semigroups that can be defined on ...
Joe's user avatar
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4 votes
1 answer
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Improve exposition of this proof: Matrix multiplication is associative, due to commutativity of underlying field

I'd like to request tips on improving proof writing by taking a standard proof in linear in algebra which is nonetheless difficult to write well, and asking for verification and improvements. I also ...
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Associative functions of real numbers with 1 and 0

What are the binary functions $F$ of the real numbers, possibly taking an open subset or including infinity, that have an identity element and a zero element and are associative? I know that $F:[-\...
1Rock's user avatar
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4 votes
1 answer
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What does associativity mean for orders?

I'm watching the class Category Theory for Programmers and it's said that an order (preorder, partial order, or total order) constitutes a category, and one of the conditions for this is that the ...
Ran Lottem's user avatar
2 votes
0 answers
31 views

Find functions $f(m,n)$ to make $a_m\times a_n= f(m,n)a_{m+n}$ associative

Let $A$ be a free Abelian monoid generated by the elements $\{a_n| n\in\mathbb{Z},n\geq 0\}$, i.e. a generic element of $A$ is a formal (finite) linear combination of $\{a_n| n\in\mathbb{Z},n\geq 0\}$ ...
Lagrenge's user avatar
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1 vote
2 answers
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Associative algebras whose induced Lie algebras are reductive.

Let $(A,\cdot)$ be a finte dimensional associative algebra over $\mathbb{C}$, which is noncommutative, and $(\mathfrak{g},[\cdot,\cdot])$ be its induced Lie algebra, i.e., $\mathfrak{g}= A$ as vector ...
zhanghtam's user avatar
2 votes
1 answer
180 views

Show that $({\rm id}\otimes \Delta)\circ\Delta=(\Delta\otimes{\rm id})\circ\Delta$ "translates" to associativity of linear algebraic groups

This is part of Exercise 2.1.3(1) of Springer's book, "Linear Algebraic Groups (Second Edition)". According to this Approach0 search, it is new to MSE. Please do not use Hopf algebras. The ...
Shaun's user avatar
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Showing associativity of gcd using a floor sum

one can express the gcd of two natural numbers using Pick´s theorem via $$ gcd(a,b) = a - b - ab + 2 \sum_{k=1}^{a} \lfloor \frac{b}{a} k \rfloor $$ I wonder how to proof the associativity. It becomes ...
Easy Mathematics's user avatar
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0 answers
68 views

Intuition of associativity of composition of functions

I am planning on taking Abstract Algebra this upcoming semester, and I wanted to read up somewhat ahead of time. Unfortunately, it seems that I am getting stuck on what should be a rather elementary ...
Daanyal Ali Akhtar's user avatar
0 votes
1 answer
41 views

Converse to a proposition regarding associative and switchable binary operations.

I define the switchability property of binary operations as follows: An ordered pair $(+,*)$ of binary operations on a set $S$ is said to satisfy switchability if for all $x,y,z$ in $S$, $(x+y)*z=x+(y*...
user107952's user avatar
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2 votes
1 answer
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Inverse Element in the Incidence Algebra of a Poset

Background I'm currently working with incidence algebras on posets. An incidence algebra was defined as $\{\text{Interval functions on } P\}$ but this notion is the same as defining the algebra as $$\...
Ignacio Rojas's user avatar
3 votes
1 answer
142 views

Check for associativity from truth table

Is there a way of checking if a binary boolean operator is associative just by looking at its truth table, similar to how you can check to see if the middle two rows are different to check if it is ...
OctopuSS7's user avatar
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2 answers
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Associativity of addition

This question is about addition of positive integers with more than one digit, a topic covered in second grade. But it's not really elementary. But first we need to introduce the positive integers $\...
Math101's user avatar
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Why is it that $\frac{d}{dx}((a+2bx)e^{3x}) \neq \frac{d}{dx}((ae^{3x}+2bxe^{3x}))$

I was computing a differential equation and I ended up with the following result: $\frac{d}{dx}((a+2bx)e^{3x})$. My intuition was to distribute $e^{3x}$, however I got the wrong answer. When I wrote ...
Peter Petigru's user avatar
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0 answers
76 views

Prove associative property of $\Bbb Z [x]$ ($\Bbb Z [x]$ is the set of all polynomials with variable x and integer coefficients)

I can't seem to find the answer anywhere to make sure the steps I did are right or wrong, and excuse me if the question is kinda dumb :") I want to prove that $\Bbb Z [x]$ is a ring, so I want to ...
Ayaznt's user avatar
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1 vote
0 answers
61 views

How to prove that (R*, x) is associative? [closed]

I understand that (R*, x) (under multiplication) is a group, where R* is the real numbers excluding 0. It has an identity element (1) Has an inverse (1/x for all x in the group) but I am unsure how to ...
Zaid Fanek's user avatar
0 votes
1 answer
87 views

If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative? [closed]

If $(\mathbb Z,*)$ is invertible and has cancellation property, does it imply that is associative? I know that if is invertible and has associative, then is a group and has cancellation property. But ...
Annai's user avatar
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1 vote
1 answer
92 views

What is the correct reading of $A+B+C$?

We have many operations where when we write it in a form that does not make it explicit which expressions are the inputs to which operands, for example $1-1-1$ is a form. Is there a name for this kind ...
user avatar
2 votes
1 answer
123 views

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 $x \in X$....
Geoffrey Trang's user avatar
1 vote
2 answers
199 views

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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