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 there a geometric proof for distributivity of integer addition/multiplication and other similar properties?

I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for ...
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Is $*$ associative?

Suppose $*$ is a binary operator on a set $A$ such that $\forall x,y\in A,$ we have $$x*(x*y)=y$$ and $$(y*x)*x=y.$$ Is $*$ associative? I can show that $*$ is commutative, because each of the ...
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Associativity of infinite matrix product.

Many texts reads "It is well known that for infinite matrices multiplication is non-associative". A treatise on this can be found in On the associativity of infinite matrix multiplication. However, ...
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Does the General Associativity Law follow from the Basic Associativity Law in the following General context? (Elementary Axiomatic Treatment)

Associative and commutative laws seem to permeate many areas. Hence, a general treatment may be of some use. In order to motivate the theory (which is necessary to even state the question), consider ...
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How many associative functions are there? [duplicate]

Suppose $A$ is a set, $f:A^2 \mapsto A$. We call $f$ associative iff $\forall x, y, z \in A$ $f(x, f(y, z)) = f(f(x, y), z)$. Now, suppose $|A| = n$. How many associative functions are there on $A$? ...
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Proof of associativity for elliptic curve point addition: Trouble understanding lines intersecting ellipic curves lemma

From Elliptic Curves Number Theory and Cryptography by Washington: $P_K^1$ is the 1-dimensional projective space. Lemma 2.2: Let $G(u,v)$ be a nonzero homogeneous polynomial and let $(u_0:v_0) \in ...
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$\Omega X$-modules are functors from $X$

Let $X$ be a connected (nice) space, $x\in X$ and $\Omega X$ the loopspace at $x$. Then $\Omega X$ has an $E_1$-structure, and so we may consider left $\Omega X$-modules. There are a few ways to do ...
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A question about the associative law of matrix

Acoording to the associative law of the matrix, the following equation $\mathbf{1}^H\mathbf{D_0}^{-1}\mathbf{D_0}^{-H}\mathbf{1} =\mathbf{1}^H(\mathbf{D_0}^{-1}\mathbf{D_0}^{-H})\mathbf{1} $, where $\...
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Why is matrix multiplication associative?

When I wonder why the product of elements of symmetric group of order 3 are associative, I just say they are isomorphic to permutation matrices and they just share that feature. Don't take me light......
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For any natural numbers $a,b,c$, prove the associativity property $(a + b) + c = a + (b + c)$.

For any natural numbers $a,b,c$, we have $(a + b) + c = a + (b + c)$. MY ATTEMPT We shall prove it by induction on $c$. For $c = 0$, we have that $(a + b) + 0 = a + b$ and $a + (b + 0) = a + b$. ...
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Growth in loops vs. growth in groups

Let $s_1,\ldots,s_k $ are elements of $G$. Let $\langle s_1,\ldots, s_k \rangle$ be the set of elements generated by $\{s_1,\ldots,s_k\}$. Let $s_{k+1} \notin \langle s_1,\ldots, s_k \rangle$. Then it ...
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Proving pair consisting of a set and binary operation is a group and whether it is Abelian.

This question is in relation to Group Theory. I am trying to determine which of the following pairs consisting of a set and a binary operation ($G$, *), is a group. And which are Abelian groups. $1....
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What are some various ways to turn $\mathbb R$ into a structure that fullfills commutative and associative laws?

For example, if addition is defined as usual and multiplication as $*(x,y)=x^ny^n$ where $n \in \mathbb N$ then obviously addition is commutative and associative and multiplication is also obviously ...
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show that the map of the multiplicative group of nonzero elements in $F$, is an isomorphism of groups. [duplicate]

Let $F$ be a field (you might assume that either $F=\Bbb R$, $\Bbb C$ or $\Bbb Z/p\Bbb Z$ where $p$ is a prime). Define the group $C$ as follows. As a set we have $C = \{(x, y)\mid x, y ∈ F,\; x^2 + y^...
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Prove $T\left(x,T(y,z)\right)=T(T(x,y),z)$

Given $$T_m(x,y)=\min(x,y),$$ for all $x,y\in[0,1]$. Prove $T_m\left(x,T_m(y,z)\right)=T_m(T_m(x,y),z)$. \begin{align*} T_m\left(x,T_m(y,z)\right)&=\min(x,T_m(y,z))\\ &=\min(x,\min(y,z)) ...
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what is this property of a multivariate function called

Given a multivariate function $f(x), x \in \Re^n$ for any $n$ and any partition of $x = \bigcup\limits_{i=1}^{m}x_i$, I need $f(x) = f(x_1, f(x_2, f(x_3, \ldots f(x_m)\ldots)))$ In essence, I have a ...
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What are $E_1$ spaces?

In one of the answers to this question, $E_1$ spaces are mentioned. In that context, they are described as a homotopically invariant version of topological group. To me, this would imply that we have ...
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In the ring $\mathbb Z_5[X]$ find associated elements with $X^3+4X^2+3X+2$ [duplicate]

In the ring $\mathbb Z_5[X]$ find associated elements with $X^3+4X^2+3X+2$ I know that I must find $a \in \mathbb Z_5[X]$ such that $a|(X^3+4X^2+3X+2)$ and $(X^3+4X^2+3X+2)|a$. Assume that this $a$ ...
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Prob. 19, Sec. 2.3, in Herstein's TOPICS IN ALGEBRA, 2nd ed: Bracketing any $n$-tuple of elements in a set with an associative binary operation [duplicate]

Here is Prob. 19, Sec. 2.3, in the book Topics in Algebra by I.N. Herstein, 2nd edition: If $S$ is a set closed under an associative operation, prove that no matter how you bracket $a_1a_2 \ldots ...
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Expressing associativity with only two variables

I'm wondering if it is possible to axiomatize associativity using a set of equations in only two variables. Suppose we have a signature consisting of one binary operation $\cdot$. Is it possible to ...
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Define $f\colon\mathbb R\times \mathbb R\to\mathbb R$ by $f(a,b)=a-b$. Is $f$ associative?

I have 3 questions about this question. First of all I'm confused about the notation. My understanding is that the first part( $f\colon\mathbb R\times \mathbb R\to\mathbb R$) means that for any ...
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Magma $(\mathbb R,*)$ with a binary operation $\;a*b=a+b-2a^2b^2$

Let $(\mathbb R, *)$ be a magma with a binary operation: $$a*b=a+b-2a^2b^2$$ Prove $(a)$ the binary operation is commutative, but not associative, $(b)$ $0$ is a neutral element for that ...
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Associative binary operation with several left inverse elememts

I am supposed to find a binary operation that is associative and has several left inverse elements. I have no clue how to do that or if it is even possible, please help :)
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Let $*$ be the operation $a*b = ab^2$ over the integers. Is this a group?

I'm trying to prove associativity and I think this operation fails. This is what I have: $$\begin{align} (a*b)*c &= a*(b*c)\\ (ab^2)*c &= a*(bc^2)\\ (ab^2)(c^2) &= (a)(bc^2)^2\\ ab^2c^2 ...
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Finite-dimensional associative algebra over an algebraically closed field

Let $V = (V, +, \cdot, *)$ be an $n$-dimensional associative algebra over an algebraically closed field $F$. Let $\{v_1, \dots, v_n\}$ be a basis for $V$. I am asked to show that, for each $i = 1, \...
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How do the coherence conditions for a monoidal category imply “associativity of the monoidal product”

Intuitively, I would say that "associativity of the monoidal product" should mean: for all objects $A,B,C$, there is a natural isomorphism so that $(A\ast B)\ast C \cong A\ast (B\ast C)$, and for all ...
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Valid reason to prove the dis-associativity of ($\mathbb R, -)$.

Is this a valid proof? Let $a,b,c$ $\in$ $(\mathbb R, -)$. We want to prove that $(\mathbb R, -)$ does not have an associative property. Assume that it is associative such that: $a-(b-c) = (a-b)-c$. $...
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Associative and Commutative

Determine which of the following operations are associative. Determine which are commutative. (a) Operation of * on Z (integer) defined by a∗b=a−b. (b) Operation of * on R (real numbers) defined by ...
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What is the reasoning behind changing the order of summation in this example?

I stumbled upon this while reading a matrix multiplication associativity proof $A(BC)=(AB)C$. I don't understand how you can change the order of summation thus can't understand the proof. I attached ...
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Is there a non-associative muliplicative closed set, with two-sided inverses and a two-sided identity?

To define a group $(G,\cdot)$ one can use the requirements: Closure Associativity A (two-sided) identity element such that $g\cdot e = e\cdot g = g$ A (two-sided) inverse for each g such that $g\cdot ...
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Showing Associativity and Commutativity of a binary operation given by a Cayley table

Let $*$ be a binary operation on the set $S:=\{0,1\}$ given by the following Cayley table: \begin{array}{c|cc} * & 0 & 1\\\hline 0 & 0 & 1\\ 1 & 1 & 0 \end{array} If I wish to ...
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Proving this non-empty set and binary operation is a group [duplicate]

Suppose we have a non-empty set $P$ equipped with an associative binary operation $\bullet$ such that for every $a \in P$ there exists a unique $b \in P$ with $aba=a$. How would we go about proving ...
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Consequence of the associative law of multiplication

Everyone knows that a * (b * c) = (a * b) * c But how to prove the same rule for more factors? I know how to do this for 4, 5 factors separately. But how to prove right away that any arrangement of ...
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how to prove group or not using associative law

Let G=(a,b)*(c,d)=(a+bc,bd), how to prove this is associative or not .i want to prove this is a group using associative law
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Are there $f:\mathbb{R}^{+}\times\mathbb R^+\rightarrow\mathbb{R}^{+}$ associative and not conjugate to the addition?

We say that Definition A function $f:X\times X\rightarrow X$ is associative if $\forall a,b,c \in X\; f(f(a,b),c)=f(a,f(b,c))$ Definition Associative functions $f,g:X\times X\rightarrow X$ ...
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How to understand a equivalent about the concept of a twisted associative algebra?

I am now reading the book ``Algebraic Operads'' written by Murray R. Bremner and Vladimir Dotsenko. In this book, on page 114-115, Definition Let $S_n$ be a permutation group. A twisted associative ...
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Multiplication, Division and Parenthesis

Considering the arithmetic expression: $8/2(4)$ Is the answer simplified to be: $(8) / (2*4) = 1$ or is it: $(8/2) * (4) = 16$? Please, explain why with your answer. Thanks!
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What does the plot/graph of an associative function on the unit square look like?

I have a function $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$. Let's say that I plot the function as a heat map (that is, I color each pixel in the square $[0,1]\times[0,1]$ according to the value of ...
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Application of associative property in the proof of finding the solution of the equation ax=b.

I am trying to find the solution for the equation $ax=b$ where $a,b \in G$ and $G$ is a group with respect to the operation, multiplication. This is what I gathered from different books $$ax=b$$ pre ...
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Is there a notion of “maximally intransitive” relation, or “maximally nonassociative” operator?

Transitivity on relations $R\subseteq X\times X$ and associativity on binary operators $+:X\times X \to X$ are defined as: $$\forall x,y,z, \quad xRy\land yRz\to xRz$$ $$\forall x,y,z, \quad (x+y)+z=...
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Artin, Proposition 2.1.4: proof of associative law of composition

This is Artin Proposition 1.4 (from his book Algebra) Suppose there is a associative law of composition given on a set S. There is a unique way to define, for every integer n, a product of n ...
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Associativity for non-binary operators

I have trouble finding a definition for the associativity from operators with arity higher than 2. At first I tried to think how it would be done for an operator with arity=3, and then generalize it, ...
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Associativity of graphs

I have a binary operation $\star$ on a set of graphs such that the $\star$ is an associative (so far i have tried) that is for any three graphs $A$, $B$ and $C$ $\implies (A\star B)\star C=A\star(B\...
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Associativity in RPN

An operator, say $+$, is associative if $a + (b+c)$ means the same as $(a+b) + c$. Rewriting this rule in RPN, we'd get that $+a+bc$ means the same as $++abc$. In this context, may I assert that an ...
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What does the associative property actually mean?

I look at $(a+b)+c = a+(b+c)$ for $a,b,c \in \mathbb{R}$ and think this tells me if I see three addends and two of them are in parentheses I can shift them without changing the sum. It obvious, at ...
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Intuition for why function composition associative

It is easy to prove that function composition is associative. I think of $f\circ (g\circ h)$ as applying first $h$, then $g$ and finally $f$ to an element of the domain of $h$. But I'm having ...
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Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
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Is category of associative algebra co-complete?

Let $R_i$ be a family of associative algebras s.t. $\phi_{ij}:R_i\to R_j$ is unital associative algebra homomorphism where I demand 1 being sent to 1 and I am assuming this is directed limit. $\...
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86 views

How to prove simple associative algebra over C is isomorphic to matrix algebra M_n(C)?

This is a problem our algebra teacher left for us. After researching on related topic, I have found out that it is a direct corollary of Wedderburn's Theorem, which is reads as follows: Suppose A is a ...
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When can the order of boolean operations be exchanged? Is there an easier way?

Suppose that we have two binary boolean operators $\circ$ and $\star$ (not necessarily different). Now in general $(A\circ B) \star C = A \circ (B\star C)$ doesn't hold, but certainly sometimes it ...

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