# 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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### 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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### 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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### 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 ...
### 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 ...