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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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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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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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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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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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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Name for operation with property x # x = x

Like, for example, min has this property because $min(x,x) = x$ Also, is there another, more specific name if this operation is also associative and commutative
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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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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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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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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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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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Does distributivity implies commutativity of one operation

Suppose there is a set $S$, equipped with two binary operations, $*$ and $@$, such that S is closed and associative under both the operations. There exist inverses and identity with respect to both ...
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Proving that a groupoid is a group, knowing the following properties

(G, ·) is a groupoid. Prove that if it has the following properties it is also a group: $$1) (a · b) · c = a · (b · c), (\forall)a, b, c \in G;$$ $$2) (∃)u ∈ G : u · a = a · u = a, (∀)a ∈ G;$$ $$3) (∀)...
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Is there a deeper reason why exponentiation is not associative?

Addition can be thought of as repeated counting; multiplication can be thought of as repeated addition; and exponentiation can be thought of as repeated multiplication. And yet, while the first three ...
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How many equivalent expressions can be derived from the associative law for addition for n summands?

The associative law for addition states: $a+(b+c)=(a+b)+c$ Considering only this law for the sum $a_{1}+...+a_{n}$ of $n$ numbers $a_{1}+...+a_{n}$. How many equivalent expressions exist? Some ...
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Existence of a neutral element in set with an associative internal law

Let $E$ be a set with an associative internal law $\star$ (i. e. such that $\star : (x, y) \in E^2 \mapsto x \star y \in E$ and for all $(x, y, z) \in E^3, (x \star y) \star z = x \star (y \star z)$). ...
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Understanding Algebras With Alteratives to the Distributive Law

If we want to quantify how much an operation $*$ associates under repeated application, we can consider the associator $(a,b,c)_* := (a*b)*c-a*(b*c)$. I understand the distributive law as a bit like ...
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If not associative, then what?

Consider a binary operation $*$ acting from a set $X$ to itself. It's useful and standard to work with operations which are associative, such that $(a*b)*c = a*(b*c)$. What about operations which are ...
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Associativity of 3d convolution in matrix multiplication format

I'm trying to understand if a 3D convolution of the sort performed in a conv layer of a cnn is associative. Specifically, is the following true:                                                         ...
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How can I show a the Cartan subalgebra being abelian implies its adjoint representation on the original Lie algebra completely commutes?

I am reading a set of notes on the classification of simple Lie algebras, in which the claim is made that since $\mathfrak h\subset g$, a Cartan subalgebra is abelian it then follows that the adjoint ...
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Is it possible to prove that $\dagger$ is associative?

Let $\dagger$ be a binary operation acted on two sets. It has the properties $$\forall A\forall B\forall C \Big(A\dagger B=A\dagger C\iff B=C\Big)$$ and $$\forall A\forall B\Big(A\dagger B=\Big((A\cap ...
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Proof that addition of subspaces of a vector space $V$ is associative

I am studying the Linear Algebra Done Right book by Sheldon Axler and one of the exercises is to verify whether addition on the subspaces of a vector space $V$ is associative. The exact phrasing of ...
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Associativity of convolution in $\ell^1(S)$, where $S$ is a semigroup.

This is what I have done but I'm not sure if it is sufficiently rigorous. Let $f,g,h\in\ell^1(S),t\in S$. Then $$((f\star g)\star h)(t)=\sum_{rs=t}(\sum_{uv=r}f(u)g(v))h(s)=\sum_{rs=t}\sum_{uv=r}f(u)g(...
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Are all alternative magmas flexible?

A magma is alternative if for all elements $x$ and $y$, we have $(xx)y = x(xy)$ and $y(xx) = (yx)x$. A magma is flexible if for all elements $x$ and $y$ we have $x(yx) = (xy)x$. Both of these are ...
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Polynomial multiplication is associative

Let $A$ be a ring and $f,g,h\in A[X]$. I want to show that $(fg)h=f(gh)$, where $(xy)_n:=\sum_{n=j+k}x_jy_k$ for all $n\in\mathbb{N}$. Attempt: Let $n\in\mathbb{N}$. I have to show that $((fg)h)_n=(f(...
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Whar are the differences between Poisson algebras with associative and Lie algebras?

Definition of Poisson algebras can be found here. The tensor algebra of a Lie algebra has a Poisson algebra structure. Moreover, by imposing an associative algebra a commutator, it turns it into ...
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When would one use Gyrovectors instead of elements of the Lorentz group to describe hyperbolic space?

Furthermore, is there a straightforward mapping between the two concepts? For example a gyrogroup homomorphism from one to the other, and a good intuition for why they fail to be associative and ...
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Saracino Ex 3.17: Associative binary operation with unique right identity, left inverse implies group [duplicate]

Suppose $G$ is a set and $\star$ is an associative binary operation on $G$ such that there is a unique right identity element and every element has a left inverse. Prove that $(G,\star)$ is a group. ...
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A more efficient proof of associativity law for a given binary operation

Consider the following definition for a binary operation on the real numbers. Let $x$ and $y$ be real numbers. Define the operation $*$ as follows. $$x * y = \begin{cases} x, & \text{if} & x \...
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Counter example for associativity of partially ordered sets

I wasn’t convinced about the associativity property of posets, and the proof i found on math.stackexchange seemed reasonable, but still i couldnt wrap my head around it. I tried to make a counter ...
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Associativity of convolution for formal power series over a ring

Let $A\ne\{0\}$ be a ring with unity and $p,q,r\in A^{\mathbf{N}}$. I want to show that $$\sum_{j=0}^n\sum_{k=0}^jp_kq_{j-k}r_{n-j}=\sum_{k=0}^n\sum_{j=k}^np_kq_{j-k}r_{n-j}$$ for all $n\in\mathbf{N}$....
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Associativity of $m * n = m+(-1)^m n$ in $\mathbb{Z}$

I have to show that $\mathbb{Z}$ with the operation $m*n = m+(-1)^m n$ is a group (Exercise 1.2.4 from Beardon's Algebra nad Geometry). There must be something wrong with this "proof" of ...
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Are there “tri-commutative” structures for which: $AB \neq BA$, $BC \neq CB$, yet $ABC = BAC = ACB$?

Groups can be Abelian or non-Abelian, however I'm curious of the space between these two where there are either weaker forms of commutativity or special properties some elements have which endows the ...
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Monoidal category: Inequivalent associators

Context In my lecture notes on tensor categories it says: "For a given category $C$ and a given tensor product $\otimes$, inequivalent associators can exist." Questions What notion of ...
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Why $(b \rightharpoonup c ) = b\cdot c$ when $b,c \in A$?

Let $A$ be an algebra. We define $\forall f \in A^*, a \in A $ $$ a \rightharpoonup f: A \rightarrow \mathbb{k}, l \mapsto f(la) $$ This operation defines a module structure on $A^*$ so it is ...
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Associativity indeed imply closure of binary operations… or what is wrong?

Similar questions have been asked without great success to answer. I read what I could about the problem, but no idea. I wrote to university professor, no response. There has been question about ...
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How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define abstract algebra)?

In elementary school, I remember learning about the basic algebraic properties of the integers like identities, commutativity, associativity, and distributivity, and not really thinking much about ...
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Grouping 2D convolutions in cascaded

Updating: When I use valid or fully padding modes I've got the same result applying associative property, the problem is when I use the zero padding mode. The edge of the convoluted image is different ...
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Can there exist a non-associative operation with an identity element,such that every element has an inverse?

As far as my understanding goes, the inverse of each element need not be even unique based on what is given. But it looks like they are asking for a where it is actually unique despite it needing not ...
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Let $(G,\cdot)$ be a set with an associative operation. Show that the following two Axioms are equivalent

Let $(G,\cdot)$ be a set with an associative operation. Show that the following two Axioms are equivalent: (a) : there exists a left-hand neutral element $e'$, so that $\forall a \in G: e'a=a$ (b): ...
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Associativity and neutral element of the composition $x \circ y=x\sqrt{y^2+1}+y\sqrt{x^2+1}$

I have to demonstrate that this composition $x \circ y=x\sqrt{y^2+1}+y\sqrt{x^2+1}$ is a commutative group, but I can't demonstrate that it is associative, and I can't find the neutral element $e$.
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Associative Law for infinitely many sets

I know the associative law for (union or intersection) of two sets and I know why it works. Applying this rule many times, will show us intuitively that this rule also holds for infinitely many sets. ...
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Associative Law - Number Sequence [closed]

I'm stuck in a problem, need Help. Associative property States that (a + b ) + c = a + (b + c) Which is true but what if i change the position of the numbers ...
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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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