# Questions tagged [associativity]

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.

267 questions
Filter by
Sorted by
Tagged with
24 views

### 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 ...
78 views

### 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 ...
20 views

### 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, ...
59 views

### 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 ...
30 views

### 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 $A$? ...
25 views

102 views

### 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......
69 views

### 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 $a + (b + 0) = a + b$. ...
29 views

### 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 ...
20 views

29 views

### 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)) ...
28 views

### 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 ...
36 views

### 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 ...
46 views

### 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 $a$ ...
48 views

29 views

### 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 ...
114 views

### 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 ...
60 views

### 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 ...
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 ...