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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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 ...
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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 $...
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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:
.
$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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Showing a non-unimodular group operation is not associative
There is a binary operation defined by
$$(f*g)(x)= \int_G f(xy^{-1})g(y)dy$$
where G is not a unimodular group.
Show is operation is not associative.
Workings so far:
$$((f*g)*h)(x)=\int (f*g)(xy^{-1}...
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Associative functions of real numbers with 1 and 0
What are the binary functions $F$ of the real numbers, possibly taking an open subset or including infinity, that have an identity element and a zero element and are associative? I know that $F:[-\...
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What does associativity mean for orders?
I'm watching the class Category Theory for Programmers and it's said that an order (preorder, partial order, or total order) constitutes a category, and one of the conditions for this is that the ...
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Find functions $f(m,n)$ to make $a_m\times a_n= f(m,n)a_{m+n}$ associative
Let $A$ be a free Abelian monoid generated by the elements $\{a_n| n\in\mathbb{Z},n\geq 0\}$, i.e. a generic element of $A$ is a formal (finite) linear combination of $\{a_n| n\in\mathbb{Z},n\geq 0\}$ ...
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Associative algebras whose induced Lie algebras are reductive.
Let $(A,\cdot)$ be a finte dimensional associative algebra over $\mathbb{C}$, which is noncommutative, and $(\mathfrak{g},[\cdot,\cdot])$ be its induced Lie algebra, i.e., $\mathfrak{g}= A$ as vector ...
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Show that $({\rm id}\otimes \Delta)\circ\Delta=(\Delta\otimes{\rm id})\circ\Delta$ "translates" to associativity of linear algebraic groups
This is part of Exercise 2.1.3(1) of Springer's book, "Linear Algebraic Groups (Second Edition)". According to this Approach0 search, it is new to MSE.
Please do not use Hopf algebras.
The ...
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Showing associativity of gcd using a floor sum
one can express the gcd of two natural numbers using Pick´s theorem via
$$ gcd(a,b) = a - b - ab + 2 \sum_{k=1}^{a} \lfloor \frac{b}{a} k \rfloor $$
I wonder how to proof the associativity. It becomes ...
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Intuition of associativity of composition of functions
I am planning on taking Abstract Algebra this upcoming semester, and I wanted to read up somewhat ahead of time. Unfortunately, it seems that I am getting stuck on what should be a rather elementary ...
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Converse to a proposition regarding associative and switchable binary operations.
I define the switchability property of binary operations as follows: An ordered pair $(+,*)$ of binary operations on a set $S$ is said to satisfy switchability if for all $x,y,z$ in $S$, $(x+y)*z=x+(y*...
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Inverse Element in the Incidence Algebra of a Poset
Background
I'm currently working with incidence algebras on posets. An incidence algebra was defined as $\{\text{Interval functions on } P\}$ but this notion is the same as defining the algebra as
$$\...
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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 ...
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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 $\...
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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 ...
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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 ...
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How to prove that (R*, x) is associative?
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 ...
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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 ...
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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 ...
user1047679
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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,
$$ (S_1 = x_1 = (x_1) ) \...
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How many associative binary operations on the integers does $+$ distribute over?
I am interested in binary operations $\mid: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ which satisfy:
Associativity: $a \mid (b \mid c) = (a \mid b) \mid c$
$+$ distributes over $\mid$: $(a \mid b) ...
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Prove that $a + (b+(c+d)) = (a+b) + (c+d)$
I'm working through Spivak's Calculus and the first section is covering associativity of addition. I am only given $P1: a + (b + c) = (a + b) + c$. I obviously know this intuitively but I cannot make ...
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Show that if $f$ is a homomorphism then the set of invertible elements $M^\times$ is commutative
I have to show the following.
Let $M$ be a monoid. If $M \to M$, $f: a \mapsto a^2$ is a homomorphism, then $M^\times$ is commutative.
So, if $f$ is a homomorphism, then for all $a,b \in M^\times$ we ...
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Field theory and distributive law.
How do any subset of a Field inherit the properties-commutativity,associativity and specially Distributive law over the same operationsas that of field?Is there any intuitive way to understand that ...
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How can I prove the associative property of this set?
Let $(G,+)$ be a group.
Let $a$ belong to $G$.
Let $C(a)=\{ g \in G \mid g+a=a+g\}$.
Does $C(a)$ satisfy the associative property?
WARNING (1) : It is not valid to prove that $C(a)$ is a subgroup. ...
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What constraints does associativity put on a real function $f(x, y)$?
We are familiar with the algebraic meaning of associativity.
But suppose we have a real function $f(x, y)$.
What constraints does
$$
f(f(x,y),z) = f(x,f(y,z))
\label{1}\tag{1}
$$
put on $f$?
Can they ...
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Is it necessary to check every possible triple of values to confirm a binary operation is associative?
Is it possible to use less than n^3 (where n is the cardinality of the set the operation works on)? That is, is there some way ...
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What is the mistake with this proof that commutativity implies associativity?
I tried to show that commutativity of addition implies associativity. For this I assumed that there is no associative property and $ a + b + c $ should be interpreted as $(a + b) + c $ .
$$(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
$$\gcd\bigg(e,\gcd(f,g)\bigg)=\...
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Is Generalized version of associative law in Halmos's set theory superfluous? How can we write a generalized commutative law?
In Halmos's naive set theory, section 9 (families), page 35 there is a part about generalized version of the associative law. It reads as follows:
The algebraic laws satisfied by the operation of ...
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Associative Law - total number of combinations satisfying the associative law [duplicate]
I'm stuck in a problem and need help. Associative law states that
(a + b) + c = a + (b + c)
If there are 4 elements, there are 5 arrangements satisfying the property.
(a + b) + (c + d)
((a + b) + c) + ...
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If the associative law doesn't hold, can we define $a^n$? [duplicate]
I am reading "Higher Algebra" by A. Kurosh.
The following sentence is in this book:
Analogously, the associative law of addition leads to the
concept of a multiple, $na$, of the element $a$ ...
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Associahedron, but with swaps
The associahedron has edges of the form $a(bc)\rightarrow (ab)c.$ But I also want to include the possibility of swapping adjacent entries by doing operations like $a(bc) \rightarrow a(cb).$ I was ...
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How do I check that this composition is associative?
I have the following problem:
Let $R$ be a ring, and $M$ a monoid. we have the multiplication on $R[M]$ given by: $$\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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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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