Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [associativity]

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

-2
votes
0answers
35 views

Proof of closure for x * y = (x^3 + y^3)^⅓

Let * be defined on ℝ as: $$ ∀ x, y ∈ ℝ: x * y = (x^3 + y^3)^⅓ $$ I have tried to prove $(x^3 + y^2)^⅓$ is a member of ℝ to show closure, but I'm not sure how. Do I need to prove it or is it just ...
1
vote
3answers
81 views

How many associative binary operations for a set of two elements?

Suppose we have a set $S=\{a,b\}.$ Obviously, the total number of binary operations on $S$ is the number of all the pairs such that $$*(a,b)=a \, \text{or} \, b,$$ which gives $ 2 \cdot 2 \cdot 2 \...
1
vote
0answers
21 views

Is there a Hamiltonian path? (i.e. can the general associative law be solved with Graph Theory?) [duplicate]

Consider the different bracketings of the summation $$1+2+3+4+5\,.$$ I've listed them all below: $$ 1+(2+(3+(4+5)))\,,\quad (1+((2+3)+4))+5\\ 1+(2+((3+4)+5))\,,\quad (1+(2+(3+4)))+5\\ 1+((2+(3+4))+...
2
votes
2answers
53 views

Matrix Multiplication not associative when matrices are vectors?

Wikipedia states: Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only the number of columns of A equals the number of rows of B and the number of columns of B ...
7
votes
0answers
165 views

Associative, non-commutative, non-trivial, analytic binary operation

There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$. As a followup, ...
1
vote
1answer
39 views

Vector operation: add angles, multiply absolue values. How is it called?

For two complex values, multiplication amounts to adding their polar-representation angles and multiplying their absolute values. For $\mathbb{R}^n$ I could define the same rule for two vectors $\vec{...
8
votes
2answers
201 views

Associative, non-commutative, nontrivial operation on the real numbers

This MSE question asks about binary operations on the real numbers which are associative, but not commutative. Two answers are given: The operation $\circ$ defined by $x \circ y=x$. Letting $f:\...
0
votes
0answers
30 views

Find or describe all associative algebras possessing only inner derivations.

From cohomology theory we know that separables algebras $A$ possess only inner derivations. In fact, they possess only inner Derivation into every $A$-bimodule $M$ which characterize separable ...
1
vote
0answers
36 views

Generalised Binary Operation?

In algebra, we study sets with binary operations satisfying certain properties. However, sometimes that the calculation is undefined for some ordered pairs, such as $a\div 0$ for any complex number $a$...
0
votes
1answer
100 views

Associative property of convolution

Consider the following sequences: $x_{1}(n) = A$ (a constant), $x_{2}(n) = u(n)$, $x_{3}(n) =\delta(n)-\delta(n-1)$. ($\circledast$ stands for linear convolution) If I perform the operation $x_{2}\...
1
vote
2answers
34 views

example of non-associativity in the physical universe?

In a recent article[1], John Baez is quoted as making a nice point about how non-commutativity is common in the world around us, whereas non-associativity is not: “[...] while it’s very easy to ...
1
vote
2answers
79 views

How many associative binary operations are there on a 2 element set?

We can easily find commutative binary operations on a 2 element set from the truth table (if ab=ba then the operation is commutative, thus there are 8 commutative binary operations in a 2 element set)....
9
votes
3answers
327 views

How many elements have to verify the associativity property in a group?

If this is a duplicate please mark it down. We know that if $(G,\ast)$ is a group then it must verify the associative property, that is, $$\forall x,y,z\in G:\quad x\ast(y\ast z)\quad=\quad(x\ast y)\...
0
votes
1answer
60 views

Is it possible to prove this associativity of the group through the inheritance of the operation?

Prove associativity in the group $(A,\cdot)$ where $$\begin{matrix}A&=&\left\lbrace1,-1,x,-x,x^2,-x^2\right\rbrace&&\text{with}&x^3=1\end{matrix}$$ and $\cdot$ product. The ...
0
votes
3answers
54 views

Does general commutativity require associativity?

There is a theorem that if an operation is associative on 3 elements from a set, it is associative on any number of them. This property is called general associativity. Similarly, there's a theorem ...
1
vote
0answers
40 views

Group order $3$ associative property - $27$ cases.

Am a novice, & am practicing to have hold of the subject. Also, request source with such questions as examples. Need show all the $27$ cases for a group's associativity property, with three ...
2
votes
0answers
48 views

Is there a measure for how associative a binary operator is?

For some binary operation defined on a set, it is of interest whether the operation is associative or not (among, of course many other things). Is there a measure (and accompanying theory) that ...
4
votes
2answers
84 views

How can I prove this operation on domino tilings is associative?

Suppose we have two tilings of a region by dominoes $T$ and $U$. If a domino in $T$ does not correspond to a domino in $U$ (that is, two cells covered by some domino in $T$ is covered by two dominoes ...
1
vote
0answers
47 views

Prove associativity for the following space.

Let $X$ be the set of all $A\subset\mathbb Z$ that are bounded above; that is, $A\in X$ iff $A\subset\mathbb Z$ and $\exists\max A$. Define the following operation, as the sum of two such sets: $A\...
1
vote
0answers
63 views

Generalized associative law

In Naive Set Theory book of Halmos there is a statement that if $\{I_j\}$ is a family of sets with domain $J$ and $\{A_k\}$ is a family of sets with the domain $K = \bigcup_{j \in J}I_j$ then $$\...
1
vote
1answer
27 views

Proving a binary operation is not associative given a latin square

Given a Latin square how would one tell if the operation is associative without trying every combination? Or is there something to look for that would at least limit the amount of combinations I have ...
1
vote
0answers
31 views

Commutativity and associativity in Stieltjes convolution algebra

I've been trying to prove the commutativity and associativity within Stieltjes convolution algebra but haven't succeeded. In the literature it says that the convolution should be both commutative and ...
1
vote
1answer
69 views

When does commutativity imply associativity in a set of operators?

This question got me wondering: suppose $S$ is a set of operators such that for all $x, y \in S$, $x(y) \in S$. For any $x,y \in S$, we say that $x \equiv y$ iff for all $z \in S$: $$x(z) = y(z)$$ ...
27
votes
10answers
5k views

Why worry about commutativity and not associativity in The Fundamental Theorem of Arithmetic?

A common statement of The Fundamental Theorem of Arithmetic goes: Every integer greater than $1$ can be expressed as a product of powers of distinct prime numbers uniquely up to a reordering of the ...
3
votes
0answers
39 views

Finding an associative operator with two operands, from its behaviour on three operands

Suppose I have forgotten how to calculate $g(a,b)=a+b$. But I have a black-box function $f(a,b,c)=a+b+c$, which I can calculate. Presumably, $f(a,b,c)=g(a,g(b,c))=g(g(a,b),c)$ uniquely specifies $g$ ...
2
votes
2answers
44 views

On the least prime in an arithmetic progression $a + nb$ where $a,b$ are distinct primes.

Dirichlet's Theorem: there are infinitely many primes in every arithmetic progression $\{a + nd: n \geq 0\}$ for coprime $a, d$. Consider just the case where $a, d$ can take on prime values or $0$. ...
3
votes
2answers
34 views

Associative algebra without nilpotent ideals is direct sum of minimal left ideals

In the book 'Spinors, Clifford and Cayley Algebras' Hermann states that any (finite dimensional) semisimple associative algebra is the direct sum of minimal left ideals. Here, 'semisimple' is defined ...
2
votes
0answers
85 views

Is this composition associative?

Given a set $X=\{1,\dots,n\}$. In Dempster–Shafer theory a BBA is a function $m:\mathcal P(X)\to[0,1]$, with the two properties that $m(\emptyset)=0$ and $\sum_{A\subseteq X}m(A)=1$. The Dempster's ...
0
votes
1answer
210 views

What happened here at the proof of associative law for addition?

I was glancing through appendix of M Artin algebra in the integers section and here is a proof by using mathematical induction on the basis of Peano's axiom. Proof for associative law for addition: ...
2
votes
1answer
98 views

A More Symmetric Exponentiation

Exponentiation is distributive over multiplication, but it isn't commutative or associative like addition and multiplication are. Is there a binary operation that is distributive over multiplication, ...
71
votes
2answers
3k views

Is there an intuitive reason for a certain operation to be associative?

When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation $$x*y=\frac{xy}{x+y+1}$$ (defined on the positive real numbers) is associative. At first I considered ...
1
vote
1answer
61 views

Semigorup variety, hyperassociativity,idempotentunclear proof of $x^4\approx x^2$

Let $V$ be a hyperasociative semigroup variety. For hyperasociativiy see below. Then $V$ satisfies the following identity: $$x^2 \approx x^4.$$ A proof attempt is given here: If $V$ is idempotent (i.e....
0
votes
0answers
192 views

How to prove the associative property of min-plus matrix multiplication?

According to Zwick, Uri, All pairs shortest paths using bridging sets and rectangular matrix multiplication, J. ACM 49, No. 3, 289-317 (2002). ZBL1326.05157. and Min-plus matrix multiplication (...
0
votes
1answer
70 views

Up to what level can associativity be guaranteed?

My question is generated from the following question: It turns out that the inverse of product with an assumption of inverse existence is a necessary condition of associative. Then is there any set ...
2
votes
2answers
44 views

Doubt in a proof of dropping parentheses with associativity

This question is from Artin Algebra: Artin gives this proof of why we can get rid of parentheses in composition if associativity is assumed: What I can't understand is what these sentences ...
1
vote
1answer
44 views

Do we have unlimited associativity when multiplying group elements with subsets?

The Question: Suppose $G$ is a group and consider the product $r_1 r_2 r_3 \cdots r_n$, where for each $i$, $r_i$ is either an element of $G$ or a subset of $G$. Does the result depend on how we ...
0
votes
1answer
37 views

What kind of associative binary operations are there on integers

Which associative binary operations on integers do you know, besides the $+,\cdot$? (It should have an identity element) Motivation for asking this: Define $E(n) = \sum_{d|n} d \log(d)$. Then for $...
4
votes
2answers
101 views

Proving that the axioms for addition hold in $R$ — Associativity. Principles of Mathematical Analysis by Walter Rudin.

I'm studying the proof for theorem 1.19 from Principles of Mathematical Analysis by Walter Rudin (public online copy here). There exists an ordered field $R$ which has the least-upper-bound ...
1
vote
1answer
48 views

What are some ways to prove that product of $n$ elements in a group does not change its value no matter how many parentheses there are?

In the book I am reading it is written: The associative law was enunciated only for three elements, but can in fact be easily proved in its more general form, namely that, without ambiguity, $...
1
vote
1answer
47 views

Associative property of a product on $L_1$-functions

If $f,g\in L_1$ we define the product $\Box$ as $$f\Box g(x)=f(x)\int_{-\infty}^xg(y)dy + g(x)\int_{-\infty}^x f(y)dy $$ The first question is if this product has a special name. So to show that it ...
1
vote
1answer
62 views

Efficient way to show the Associativity of $\oplus: E \times E \rightarrow E,\ x\oplus y \equiv x+y \bmod 5$

Let $E:=\{0, 1, 2, 3, 4\}$ be a set and let $\oplus$ be an internal binary operation on $E$ such that $$\oplus: E \times E \rightarrow E,\ x\oplus y \equiv x+y \bmod 5$$ So, we have: \begin{array}{|...
-1
votes
1answer
49 views

Prove associative property

Given: $\mathbb{R}\times(\mathbb{R}\setminus\{0\})$ with the following operation $\circ$ $(a_1,b_1)\circ (a_2,b_2):=(a_1\cdot b_2+a_2,b_1\cdot b_2)$ I need to prove the associative property. So: $\...
0
votes
0answers
198 views

Show that subtraction of matrices is neither commutative nor associative. Please review my work.

Show that subtraction of matrices is neither commutative nor associative. My Work Let $A$ and $B$ be $m \times n$ matrices with elements $a_{ij}$ and $b_{i j}$, respectively. Commutativity says ...
2
votes
1answer
269 views

Testing for associativity using the multiplication table (Cayley Table) of an operation. [duplicate]

I seem to recall that there is a relatively easy method for determining the associativity of an operation by using its Cayley table. What is it?
-3
votes
1answer
86 views

Is there an non-linear binary operation that's associative?

In cryptography, non-linearty is an important property in symmetric primitives that ensures trivial linear analysis cannot succeed; associativity is a property that is essential in public-key (a.k.a. ...
0
votes
1answer
71 views

Understanding associative operations

If something is associative, then for example $(x + y) + z = x + (y + z)$ is true. I also know that in general subtraction is not an associative operation, but what if subtraction is applied to $\...
2
votes
3answers
74 views

Commutativity + “something” $ \to $ associativity?

What additional properties must an operation have besides commutativity so that commutativity along with other properties implies associativity? Where can I read about such structures?
2
votes
2answers
103 views

Sigma notation: what should be the equivalent associative law for product notation?

Below is the associative law for sigma notation. $$\sum_{k \in K} a_k + \sum_{k \in K} b_k = \sum_{k \in K} (a_k + b_k)$$ What should be the equivalent law for product notation ? and why ? Should ...
0
votes
1answer
104 views

XOR associative property in parity calculation

So I am trying to determine the parity of an 8 bit word. Parity means that if the number of 1's in the word is odd the parity=1 and if number of one's is even parity=0. The word I am trying to ...
2
votes
1answer
60 views

How is the phenomenon of “reverse associativity” called?

I am trying to find either a name or a way of modeling the following phenomenon: Let's use the functions $add(x, y)$ and $max(x, y)$. Both these functions are known to be associative and commutative. ...