Questions tagged [associativity]
This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.
406 questions
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Unconditional convergence and analytic map
Here is the definition of an analytic map.
ANALYTIC MAP
Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if
$$
f(x)=\sum_{k=0}^{\infty}a_k(x) \tag 1
$$
where
for ...
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Associativity of the sum in an analytic function
Here are two definitions.
ANALYTIC MAP
Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if
$$
f(x)=\sum_{k=0}^{\infty}a_k(x)
$$
where
for each $ k$, the map $a_k \...
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How does this algorithm "rank" amongst other in the literature?
By merging together the contributions from: a) this answer, b) the comments under this answer, we come up to the following:
Claim. For $n\in\mathbb N$, let $Q=(\{1,\dots,n\},*)$ be a quasigroup. Then, ...
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A transitive subgroup of order $n$ of $S_n$ inducing a non-associative latin square.
For $n\in\mathbb N$, the multiplication table of a quasigroup $Q=([n],*)$ induces a transitive subset $\Sigma=\{\sigma_1,\dots,\sigma_n\}\subseteq S_n$ via the position:
$$\sigma_i(j):=i*j\tag1$$
and ...
- 5,601
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Calculations involving absolute values - is it necessary to write down all the procedures?
I am trying to show that the operation $\ast$ gives an associative binary operation on $\mathbb{R}^*$, where $\mathbb{R}^*$ is the set of all real numbers except $0$, and $\ast$ on $\mathbb{R}^*$ is ...
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Associative algebras with ad nilpotency conditions
I'm interested in the following construction: Let $V$ be a vector space and $A(V)$ be the corresponding free associative algebra on $V$, aka the tensor algebra of $V$. Consider a two-sided ideal $I_n$ ...
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Two-variable polynomial that is associative, not commutative, and not a projection
Is there a two-variable (real) polynomial that is associative, not commutative, and not a projection?
That is: Does there exist a polynomial $p$ such that
$\forall x, y, z. p(p(x, y), z) = p(x, p(y, ...
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What are the semantical pre-requisites for an operation being associative?
I beginning to start math from the start again and I'm trying to understand things more intuitively and I'd like some help to understand associativity. I can understand it as the characteristic of an ...
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$\Omega$-algebras and their semidirect product
$\Omega$-algebras are defined as an algebra with $\Omega$ that consists of $n$-ary operations, $n \geq 2$. Is the concept of semidirect product available for them?
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Alternative proof that a power-associative magma with no idempotent elements is infinite.
I have been investigating commutative, associative magmas in an ad hoc way for the past few days and was curious about idempontent-free magmas. The magma $(\mathbb{Z}_{\ge 1}, +)$ is idempotent-free, ...
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Does there exist an associative, commutative, and bijective (up to reordering) binary operation on the natural numbers?
Here's where the motivation comes from. I was trying to figure out a way to assign a unique natural number to every pure set (that is, I wanted to find a bijective function with a domain of pure sets ...
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Group structure from unique solvability of linear equations? [duplicate]
Commonly, a group is defined as follows:
Definition. Let $G$ be a set and $\ast:G\times G\to G$ a binary operation on that set. Then the pair $(G,\ast)$ is called a $\textbf{group}$ if the following ...
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How to determine whether the following two notions of "partial associativity" are inequivalent
Definitions:
Given a triple $(S,T,\cdot)$, let us say $(S,T,\cdot)$ has property $R$ if $S$ is a set, $T \subseteq (S \times S)$ is a relation from $S$ to itself, and $\cdot : T \rightarrow S$ is a ...
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What property ensures order of operations affects the result?
I'm trying to understand a property of binary operations where the order of applying the operation to a set of elements always matters. I'm not super advanced in math, but I will try to explain what I ...
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How do you prove associativity for this operation?
Let $S$ be a set with a binary operation $/:S\times S \rightarrow S$ where $(a,b) \mapsto a/b$ such that:
There exists an element $1 \in S$, such that $a/b = 1$ if and only if $a=b$.
For any ...
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