Intereting Posts

random thought: are some infinite sets larger than other
Hellinger-Toeplitz theorem use principle of uniform boundedness
A set with a finite integral of measure zero?
Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E$ an element of the dual space of $C_c^∞(G)$?
Result of the product $0.9 \times 0.99 \times 0.999 \times …$
Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”
how to solve double integral of a min function
Difference between “Show” and “Prove”
Any N dimensional manifold as a boundary of some N+1 dimensional manifold?
Let R be a relation on set A. Prove that $R^2 \subseteq R <=>$ R is transitive $<=> R^i \subseteq R ,\forall i \geq 1$
Relative countable weak$^{\ast}$ compactness and sequences
Homotopy invariance of the Picard group
How to find the logical formula for a given truth table?
Find the volume of the set $S=\{x=(x_1,x_2,\cdots,x_n)\in \Bbb{R}^n:0\le x_1\le x_2\le \cdots \le x_n \le 1\}$
$ax+by+cz=d$ is the equation of a plane in space. Show that $d'$ is the distance from the plane to the origin.

Does commutativity imply associativity? I’m asking this because I was trying to think of structures that are commutative but non-associative but couldn’t come up with any. Are there any such examples?

NOTE: I wasn’t sure how to tag this so feel free to retag it.

- How to tell if some power of my integer matrix is the identity?
- Set of prime numbers and subrings of the rationals
- If R is a PID, is it true that $R/\ker \phi$ is also a PID?
- Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.
- Cancellative Abelian Monoids
- GCD in polynomial rings with coefficients in a field extension

- Sum and intersection of submodules
- Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$
- Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$
- A ring without the Invariant Basis Number property
- Direct limit of $\mathbb{Z}$-homomorphisms
- Maximal Ideals in the Ring of Complex Entire Functions
- Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
- Finding the Galois group over $\Bbb{Q}$.
- Number of elements of order $7$ in a group
- $I+J=R$, where $R$ is a commutative rng, prove that $IJ=I\cap J$.

Consider the operation $(x,y) \mapsto xy+1$ on the integers.

A basic example is the “midpoint” binary operation: $a*b = \frac{a+b}{2}$

In general, if $P(u,v)$ is any polynomial in two variables with rational coefficients, then $x*y = P(x+y,xy)$ is rarely associative – I’d be curious under what conditions on $P$ this operation would be associative.

My example is $P(u,v)=\frac{u}{2}$ and Marlu’s example is $P(u,v)=1+v$.

The easiest Jordan algebra is symmetric square matrices with the operation

$$ A \ast B = (AB + BA)/2, $$

similar to a Lie algebra but with a plus sign.

Let $A = \{e,x,y\}$. Define $\cdot$ on $A$ to be $a\cdot e=a$ for all $a$, $e\cdot a= a$ for all a, and $a\cdot b=e$ for all $a$ and $b$ such that $a\neq e$ and $b\neq e$, (i.e. $a,b \in \{x,y\}$).

This operation is commutative, $e$ is the identity, (everything even has an inverse), but is not associative since $(x \cdot y) \cdot y = e \cdot y = y$ and $x \cdot (y \cdot y) = x \cdot e = x$.

Arguably the most important example of a commutative but non-associative structure is that of finite-precision floating point numbers under addition. `(a + -a) + b`

is always equal to `b`

but `a + (-a + b)`

can differ from `b`

since the sum `-a + b`

can involve a loss of precision (this is especially true if `a`

and `b`

are nearly but not quite equal, `-a + b`

could work out to `0`

even though the corresponding real sum is nonzero). The lack of associativity of floating point arithmetic is a constant complicating factor in numerical analysis.

Consider the commutative operation $\texttt{vs}$ on the set $\{\textbf{rock}, \textbf{paper}, \textbf{scissors}\}$ abbreviated $\{r,p,s\}$ defined by

$$ \begin{array}{c|ccc}

\texttt{vs} & r&p&s\\ \hline

r & r & p & r \\

p & p & p & s \\

s & r & s & s

\end{array}$$

It is not associative since, for example, $$\textbf{paper} \texttt{ vs } (\textbf{scissors} \texttt{ vs } \textbf{rock}) = \textbf{paper}$$

but

$$(\textbf{paper} \texttt{ vs } \textbf{scissors}) \texttt{ vs } \textbf{rock} = \textbf{rock}.$$

The simplest examples of commutative but nonassociative operations are the NAND and NOR operations (joint denial and alternative denial) in propositional logic. To quote from my answer to another question:

Namely, the $2$-element structure $\{a,b\}$, where $aa=b$ and $ab=ba=bb=a$, is commutative but not associative. This is the unique (up to isomorphism) binary operation on a $2$-element set which is commutative but not associative; it can be interpreted as either of the truth-functions NOR or NAND.

- Connected Implies Path (Polygonally) Connected
- Good Physical Demonstrations of Abstract Mathematics
- Given an injection $\mathbb{N}\to\mathcal{P}(X)$, how can we construct a surjection $X\to\mathbb{N}$?
- the numer of partial function between two sets in combinatoric way
- Does $f^3$ integrable imply $f$ integrable?
- Under what conditions is a linear automorphism an isometry of some inner product?
- How to define addition through multiplication?
- A polynomial whose Galois group is $D_8$
- Taylor series convergence for $e^{-1/x^2}$
- How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?
- Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
- Show that a matched set of nodes forms a matroid
- Why can the intersection of infinite open sets be closed?
- Prove that $f$ is a polynomial if one of the coefficients in its Taylor expansion is 0
- How to prove $\prod _{a=0}^{9}\prod_{b=0}^{100}\prod_{c=0}^{100}(w^a+z^b+z^c)\equiv?\pmod {101}$