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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Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
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Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
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A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
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Does associativity imply commutativity?

I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ ...
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When a loop with inverse property is commutative

Question How to prove that a loop $L$ with inverse property and $x^3=e$ for all $x$ is commutative iff $(x y)^2=x^2 y^2$ for all $x,y$? Definitions: A loop is a quasigroup with identity $e$. ...
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Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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1answer
33 views

Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: $$f_n(x_0,\...
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Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)

I have recently found some interesting properties of the function/operation: $x⊕y = \frac{1}{\frac{1}{x}+\frac{1}{y}} = \frac{xy}{x+y}$ where $x,y\ne0$. and similarly, its inverse operation: $x⊖y = ...
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1answer
59 views

how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
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Is $\mathbb{Z}/n\mathbb{Z}$ a group under multiplication when $n$ is prime?

I have to prove that $\mathbb{Z}/n\mathbb{Z}$ is not a group under multiplication for all n>1. I would argue, however, that when $n$ is prime, $\mathbb{Z}/n\mathbb{Z}$ is a group under multiplication. ...
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28 views

How to prove associative law for groups

I'm having trouble figuring out the proof to the proposition: for any $a_1,a_2,\ldots,a_n \in \mathbb{G}$ the value of $a_1~R~a_2~R~a_3~R\cdots R~a_n$ is independent of how the expression is bracketed ...
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Are there interesting examples of distinct operations that associate?

I was just reading a text on first-order modal logic and found it very interesting to think of the distinction between rigid and non-rigid designators as a failure of associativity between the modal ...
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1answer
27 views

Is it right to say: A finite semigroup always contains an idempotent element? [closed]

Is there a theorem which says: If an operation on a set is associative then the set contains idempotent?
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1answer
24 views

Proof for association law?

I am new in logic and I getting a little bit confused with maths. Can I do something like this following the Associative Law? $$(p ∨ ¬r) ∨ (r ∨ ¬p) ≡ (p ∨ ¬p) ∨ (r ∨ ¬r)$$ Thank you in advance for ...
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What is the one-one and onto mapping?

I'm reading the following paper, and here I can't understand the first step of the algorithm. Please explain what is the one-one and onto mapping? And how he gets these tables ? Commutative ...
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1answer
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Associativity? Can this be applied here?

As the Associativity law says that (A ∧ B) ∧ C ≡ (A ∧ C) ∧ B, can I do something like this? (A ∧ ¬B) ∨ (B ∧ ¬A) ≡ (A ∧ ¬A) ∨ (B ∧ ¬B) I am new with logic and I still don't get this basic principles....
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Associativity of the Dirichlet Convolution Product

How can you prove that the convolution product of aritmetical functions is associative, and that it is distributive in respect to the addition? The book that i'm reading states that (F_a, ) is a ...
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1answer
17 views

Induction with associative binary operation

Let * be an associative binary operation on a set 'A' with identity element e. Let 'B' be a subset of 'A' that is closed under *. Let b1, b2, b3, ... bn ∈ B. Prove that b1 * b2 * b3... bn ∈ B. ...
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If $H$ is a subloop of a finte loop $L$ and $N$ is a normal subloop of $L$, then $HN$ is a subloop of $L$.

To prove this is a subloop, I have to show that for $x, y \in HN$, the following are also in $HN$: (a) $xy$, (b) $L^{-1}_x(y)$ and (c) $R^{-1}_{x}(y)$. Here $L_x(y) = xy$ and $R_x(y)=yx$. We have to ...
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1answer
109 views

Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
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4answers
67 views

Right way to show associativity.

Authors usually write that $*$ is associative on a set $S$ if, $(a*b)*c=a*(b*c)$ $\forall a,b,c \in S$ I think it should have been, $(a*b)*c=a*(b*c)=(a*c)*b$ $\forall a,b,c \in S$ I made all ...
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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: $ab=(q_0,{\bf{q}})(p_0,{\bf{p}})=(p_0q_0-{\bf{q}}\cdot{\bf{p}},q_0{\bf{...
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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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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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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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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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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: $x^{n+1}=x, \forall x\in S$ $xy^nx=yx^...
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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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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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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
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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 $a_1,a_2 \in K$...
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1answer
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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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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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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 left-...
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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
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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 ...
2
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1answer
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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) ...
2
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2answers
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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 ...