# Questions tagged [associativity]

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

392 questions
Filter by
Sorted by
Tagged with
154 views

• 3,773
1 vote
55 views

### Proof for associativity of a modified "matrix multiplication"

I'm trying to prove the associativity of a modified form of matrix multiplication (which is defined below) and I found the following proof which I'm confused about: For matrices $W_i, W_j, W_p$, to ...
64 views

### Why is 1+2+3 = (1+2)+3 [duplicate]

Not sure if this is a stupid question, apologize if it is. I am curious why we can add the first 2 numbers, then add the third one when doing addition of 3 numbers. There is a similar (IMHO) question, ...
• 109
1 vote
24 views

70 views

### 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 ...
• 894
1 vote
57 views

• 145
32 views

### 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 ...
1 vote
94 views

### 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 ...
• 556
93 views

• 2,115
123 views

### 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 ...
• 143
31 views

### 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 $\{a_n| n\in\mathbb{Z},n\geq 0\}$ ...
• 836
1 vote
112 views

### 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 ...
180 views

### 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 ...
• 45.7k
1 vote
46 views

### 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 ...