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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Associative set operation? [duplicate]

Define $A+B=(A\cup B) \backslash (A\cap B)$. Is this set operation associative? I've try to expand $(A+B)+C$ and $A+(B+C)$, but it just became too messy...
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Associvity and Distributive property of Matrix Multiplication

Matrix multiplication is associative $(AB)C=A(BC)$ What about the case when $AB$ Results in a scalar? Consider the case when $A$ is $1 \times n$ dimenional, $B$ and $C$ are both $n \times 1$ ...
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How to prove associativity of quaternion multiplication using scalar and vector form?

In scalar and vector form, a quaternion can be represented as $a=(q_0,{\bf{q}})$. The definition of quaternion multiplication is: ...
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Proving associative property, floor function

I need to prove the following operation is associative: $x*y = xy \pmod 5$ I came up with the equation that $x*y=xy-5[\![xy/5]\!]$ I'm having difficulty proving that $x*(yz)=(xy)*z$. After ...
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Direct product, semidirect product and associativity

I know that the direct product operation is associative, and that in general the semidirect product operation is not associative. But what about when we work with a mix of both? I'm trying to ...
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Non-associative commutative binary operation [duplicate]

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative ...
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Expressing associativity as a Boolean SAT problem

Suppose we have a binary operation $B: S \times S \to S$ on a finite set $S$. We can encode $B$ as an array of $|S|^3$ Boolean values by setting $$ B_{ijk} \equiv \text{True} \quad \text{ iff } \quad ...
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74 views

Anti-associativity and a sign problem

I'm doing some algebraic manipulations, and I'm getting crazy over a stupid sign error. I think I've located the source of the problem. It should come from an error I'm making (but can't see) in the ...
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20 views

Inductive proof of associativity of free groups

I'm really struggling with the inductive proof of the associativity of free groups, given about halfway down page 6 of this pdf. The bit I'm not getting is this: Suppose now that bc involves a ...
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6answers
77 views

Associative Property

What I learn as a basic theory of associative is that $$(a \times b) \times c = a \times (b \times c) \text{ and } (a+b)+c = a+(b+c).$$ However when doing my exercises on this topic, I came across ...
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jacobi identity implies flexible algebra

I was looking in the wikipedia article about Non-associative Algebra and came across this interesting line: Each of the properties associative, commutative, anticommutative, Jordan identity, and ...
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52 views

Associative operation on a set $S$

Let $S$ be a non-empty set and $n\in \mathbb{N}, n\geq 2$ a fixed integer. We consider an associative operation $"\cdot"$ on $S$ with the following properties: $x^{n+1}=x, \forall x\in S$ ...
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Meaning of closed under an associative product

I know the meaning of associative binary operation. I know the meaning of closed under a binary operation. Does closed under an associative product means: $$ a*(b*c) \in G \implies (a*b)*c \in G$$
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The Pontrjagin product for homology at the chain level

The singular cochain complex of a space $X$ with coefficients in a ring $R$ can be endowed with a product, turning it into a differential graded $R$-algebra, such that in cohomology it gives the ...
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140 views

Non-associative: set with a binary operation, but has inverses and identity

I've been thinking about an example of some set with a binary operation which would satisfy all axioms of groups except for associativity. I'm new to Group Theory, so I would appreciate your ...
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Binding two databses of association rules?

Assuming we have $D_1$ and $D_2$ as two databases of same domain, but with different Item sets $T_1$ and $T_2$. My question is that if there is any lemma to bind or ...
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25 views

Does the “necessarily” part in the question make a difference? [duplicate]

"Given an associative F-algebra do there necessarily exist elements a1.a2∈K satisfying a1∗a2=a?" I have a question to add to the one linked below: Given an associative $F$-algebra do there ...
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1answer
68 views

Given an associative $F$-algebra do there necessarily exist elements $a_1,a_2 \in K$ satisfying $a_1*a_2=a$?

Let $F$ be a field and let $(K,*)$ be an associative $F$-algebra which, as a vector space, is finitely generated over $F$. Given an element $a\in K$, do there necessarily exist elements $a_1,a_2 \in ...
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32 views

Proving two elements of a set are equal based on a two-sided identity

Say I have a set S w/ an associative binary operation *: S x S -> S and a two-sided identity e, and let . Let L and R be elements of S such that L * s = e = s * R How can I prove that L = R ? Since ...
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86 views

How do you prove the left and right side of an identity of a set are equal?

I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right ...
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1answer
12 views

Associativity of ore-polynomials

I'm fairly new to ore-polynomials. Until now I thought the rule $\frac{d}{dt}x(t)=\dot{x}(t)+x(t)\frac{d}{dt}$ resp. $x(t)\frac{d}{dt}=\frac{d}{dt}x(t)-\dot{x}(t)$ would do the trick in right- or ...
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Is the operation $x*y=xy/(x+y+1)$ associative or not?

This questions is from Charles C. Pinter's "A Book Of Abstract Algebra, Second Edition". This is the 7th question of Chapter 2 on page 24. In the selected answers section it says it is not associative ...
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1answer
131 views

Proof of associativity for concatenation operation

I recently took a mathematical proof class and am beginning to teach myself abstract algebra. I'm fairly new to proofing however, and am not very confident in how I do it. Also, I'm new to this site ...
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Confused by inductive proof of associative law

In Artin's book he proves the associativity of a $n$-element product. It says as follows: i) the product of one element is the element itself. ii) the product $a_1a_2$ is given by the law of ...
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Order of calculation in a Semigroup.

If $(A,\cdot)$ is a semigroup, i.e. if we have: $\forall (x,y,z)\in A^{3}, (x\cdot y)\cdot z=x\cdot(y\cdot z)$, then the order of calculations doesn't matter no matter the number of factors and we can ...
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Proof of associativity - Algebra

How to prove that the algebra $\langle \mathbb{Z}_6, +_6\rangle $ is associative?
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Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
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Proving associativity in Algebra

How to proof that a specially defined Transitive Join for the relations $R \subseteq A$ x $B$ und $S \subseteq B$ x $C$ is associative? The join is defined as: $R \Join S =_{def} \{(a,c)| $ there is ...
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When is convolution associative?

Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. What about the convolution of measures on an unimodular group $G$?
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Associativity of product law in $R^S$ ($R$ ring, $S$ a monoid with condition)

In Proof of associativity of polynomials product (infinite variables), I ask a question about polynomials and assume it was linked to a question of total algebra. I explicitely ask this question here ...
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Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
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Is there an example of nonassociative arithmetic addition?

Are there any clear, accepted examples of operations that are appropriately defined as "addition" but are not associative? Although I can find references to abstract discussions of arithmetic systems ...
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Semigroup where the Binary Operation is not Associative. [closed]

I am working on my functional composition, which has the associative property, to show if a given pair is a semigroup or not. I believe all Semigroups have to have a binary operation that is ...
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1answer
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Associative Proof

I have a non-empty set $R$ with a binary operation $*$. If the pair has an identity element $e\in R$ and $$(a*b)*(c*d)=(a*c)*(b*d)$$ holds for all $a,b,c,d \in R$, how do I then prove that this is ...
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Semigroup law on points on the curve $f(x) = \frac{1}{x}$

Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$. Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve. Draw a line through them Find where this point ...
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Does the following property of the composition of a magma have a name?

If $M$ is a magma and $$+:M\times M\to M$$ is its law of composition, does the property $$(x+y)+z=x+(y+z)\qquad\forall\ x,y,z\in M :\quad y\neq x,z$$ have a name? It resembles the associativity of ...
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Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
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116 views

Non-Commutativity Implies Non-Associativity?

I read this question, Does commutativity imply Associativity?. And then was curious if non-commutativity implies non-associativity. For concatenation (*), this is ...
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How does associativity work for this fuzzy norm?

How to prove that $$\min\left(1,\min(1,x+y)+z\right)=\min\left(1,x+\min(1,y+z)\right)$$ where $x,y,z\in[0,1]$?
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1answer
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Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
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Finding probability associations of two types of classes

Suppose we have k number of Objects. Every object belongs to 1 A-type class and 1 B-Type class. There are n number of A type classes and m number of B type classes. Given probabilities (total k ...
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How does one “parenthesize elements”

This is probably something very obvious, but I am a little confused. It's about the associative law. It is known that a binary structure $(S, *)$ is associative if: $(a * b) *c = a * (b * c)$ for ...
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1answer
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Proving Non-Associativity of a Binary Operation in $\mathbb{R}$

The operation $(*)$ is defined as $$a*b=|a-b|, \forall a,b \in \mathbb{R},$$ and I am to prove that $(*)$ is not associative in $\mathbb{R}$, that is, to prove that it is not true in general that ...
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What are other examples of complex associative operators besides, x + y +rxy, rxy, and x + y + 1/r?

I have been having fun (and frustration) in finding complex associative operators over the complex numbers. So far, I have found the 3 listed in the title (r is a constant), and also know about ...
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Associativity in category theory [closed]

In Category Theory, the composition of morphisms must be associative. What would happen if we give up this associative law? For example Lie algebras are vector spaces with non-associative binary ...
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17 views

Associative and anticommutative Binary Operation(composition)

Show that if binary operation ,$\Delta$, is associative and anticommutative on $\mathbb{E}$, then $x\Delta y \Delta z=x\Delta z$ ∀$x,y,z \in \mathbb{E}$. [Hint: consider $x\Delta y\Delta z\Delta ...
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Associative Binary Operation(composition) is anticommutative iff idempotent…

if Binary Operation, $\Delta$, defined on $\mathbb{E}$ is associative, then $\Delta$ is anticommutative iff $\Delta$ is idempotent and $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.
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Can we give a bound on any associative function?

We say that $f:[1,\infty)^2\to[1,\infty)$ is associative if $$f(f(a,b),c)=f(a,f(b,c))$$ And symmetric if $$f(a,b)=f(b,a)$$ e.g. the arithmetic operations '+' and '$\cdot$' are associative and ...
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277 views

Using the properties of real numbers, verify that complex numbers are associative and there exists an additive inverse

I am self-learning, so I need guidance, as I am unsure whether my approach is sufficient. There are two questions, both asking to verify a property of the complex numbers using the properties of real ...
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66 views

Semantics of operator $\times$ with regards to sets

I am trying to understand something about operator $\times$ with regards of sets. In the accepted answer to the following question (The cross product of two sets), the answerer says that $A \times B$ ...