All Questions
Tagged with magma associativity
15
questions
-4
votes
1
answer
79
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Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]
Associativity is:
$$(a * b) * c = a * (b * c)$$
Alternativity is:
$$a * (a * b) = (a * a) * b$$
$$(a * b) * b = a * (b * b)$$
Bol loop is:
$${\displaystyle a(b(ac))=(a(ba))c}$$
$${\displaystyle ((ca)b)...
- 3,773
3
votes
0
answers
213
views
Does the percentage of associative operations on a finite set decrease monotonically towards zero?
In this answer, André Nicolas proves that it is rare for a binary operation on a finite set to be associative, in the following sense: if $A_n$ denotes the number of semigroups that can be defined on ...
- 20.8k
7
votes
2
answers
466
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If not associative, then what?
Consider a binary operation $*$ acting from a set $X$ to itself. It's useful and standard to work with operations which are associative, such that $(a*b)*c = a*(b*c)$. What about operations which are ...
- 1,324
1
vote
1
answer
81
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Are all alternative magmas flexible?
A magma is alternative if for all elements $x$ and $y$, we have $(xx)y = x(xy)$ and $y(xx) = (yx)x$. A magma is flexible if for all elements $x$ and $y$ we have $x(yx) = (xy)x$. Both of these are ...
- 293
1
vote
1
answer
168
views
Magma $(\mathbb R,*)$ with a binary operation $\;a*b=a+b-2a^2b^2$
Let $(\mathbb R, *)$ be a magma with a binary operation:
$$a*b=a+b-2a^2b^2$$ Prove
$(a)$ the binary operation is commutative, but not associative,
$(b)$ $0$ is a neutral element for that ...
- 4,670
0
votes
2
answers
41
views
Valid reason to prove the dis-associativity of ($\mathbb R, -)$.
Is this a valid proof?
Let $a,b,c$ $\in$ $(\mathbb R, -)$. We want to prove that $(\mathbb R, -)$ does not have an associative property. Assume that it is associative such that: $a-(b-c) = (a-b)-c$.
$...
- 52
1
vote
0
answers
90
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Why is a "double-cancellative" operation so weird?
Let $A = \{ \{0\}, \{0,1\} \}$.
Let $\bar{A}$ be the family of sets generated by the Cartesian product on $A$.
This is a magma $(\bar{A}, \times)$ that has what I am calling a "double cancellative" ...
- 5,302
3
votes
1
answer
145
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Is power-associativity an equational property?
A magma $M$ is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as $x^mx^n=x^{m+n}$ for all $m,n$ positive integers and $x\in M$, ...
- 27.7k
2
votes
0
answers
46
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Does the following property of the composition of a magma have a name?
If $M$ is a magma and
$$+: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$$
have a name? It resembles the associativity of ...
- 588
2
votes
1
answer
1k
views
Associativity in category theory [closed]
In Category Theory, the composition of morphisms must be associative. What would happen if we give up this associative law? For example Lie algebras are vector spaces with non-associative binary ...
- 69
16
votes
6
answers
1k
views
Just How Strong is Associativity?
A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "...
- 45.8k
5
votes
2
answers
185
views
Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$
When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that
$$a*(b*c)=(a\cdot b)*c$$
If $*$ is associative then $\cdot=*$ even if I'm not sure ...
- 2,512
0
votes
1
answer
412
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Discuss whether or not the following binary operations are commutative, associative, ...
Discuss whether or not the following binary operations are commutative, associtive, have neutral elements and for which elements there are inverse elements.
In between are what I have said, but if it ...
- 111
13
votes
6
answers
2k
views
Associativity test for a magma
Say I have the operation table for a magma. I want to know whether or not the operation is associative. However, associativity is defined for an operation on 3 elements, and the operation table deals ...
- 457
62
votes
7
answers
20k
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Is there an easy way to see associativity or non-associativity from an operation's table?
Most properties of a single binary operation can be easily read of from the operation's table. For example, given
$$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e\\\hline
a &...
- 43.7k