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Questions tagged [quasigroups]

A quasigroup is a grouplike structure $(Q, \ast)$, that satisfies the Latin square property but need not have an identity element, nor need it be associative. It coincides with the notion of a divisible magma.

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Why quasi-random sequences are generated in the interval [0,1]? Is it a normalized sequence generation?

The quasi-random sequences are generated using low discrepancy sequences and Koksma-Hlawka inequality explains the quasi sequence clearly. However, it is observed that these sequences are generated in ...
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Multiplication table of a commutative quasi group is a symmetric Latin square.Does the also converse true?

Or can we find an example where the Latin square is symmetric but it's corresponding quasi group is not commutative?
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Definition of Additive Loop

I have thought of loops as having the operations of multiplication and left and right division. I read the D. R. Hughes article on Additive and Multiplicative Loops of Planar Ternary Rings and it ...
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Commutative subtraction

It is well known that subtraction is not commutative in general. However, it is commutative in some groups: $\mathbb I$, $\mathbb C_2$, $\mathbb K_4$. I am trying to understand the logic. ...
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Invertibility as Criteria for a Loop

I try to understand the correct criteria for a Loop. I see in Wikipedia https://en.wikipedia.org/wiki/Inverse_element#In_a_unital_magma that “A unital magma in which all elements are invertible is ...
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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 ...
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In a finite monoid (M, $\circ$) if the identity element $e$ is the only idempotent element, prove that each element of the monoid is invertible.

In a finite monoid (M, $\circ$) if the identity element $e$ is the only idempotent element, prove that each element of the monoid is invertible. As the set $M$ is finite, $\exists$ $y \in M$, s.t $y \...
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Congruence lattice of direct power of Steiner seven element quasigroup

The Steiner seven element quasigroup is the algebra $\mathbf{S}_7 = \langle S_7, \cdot \rangle$, where $S_7 = \{ 1,2,3,4,5,6,7 \}$, and, up to isomorphism, its multiplication table is the one given ...
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Complexity of testing if a binary operation is a group

Given a binary operation specified as an $n \times n$ Cayley table, what is the complexity of the best deterministic algorithm for testing if the binary operation is a group? There's a fairly simple ...
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Conflict with finite Moufang loop solvability propositions

I really got stuck with the following contradiction. Say we have a Moufang loop $Q$, $|Q| < \infty$. To put it briefly, Moufang loops are groups that not necessary be associative, with extra ...
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Examples of proper loops in $\mathbb{R}$

A loop $(L, \cdot)$ is a binary structure that satisfies every group axiom except for the associative property. A loop which is not a group is called a proper loop. A topological loop $(L,\cdot)$ is a ...
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Composition Law to make a Quasigroup

Here is the problem I am attempting to solve: Let $S$ be a set with a composition law $\cdot$ possessing the following properties: $$(i)~x\cdot y = y\cdot x~\forall~x,y\in S$$ $$(ii)~x\cdot (x\cdot y)...
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Prove that a map is not quasi-isometric embedding

I am not sure that I’m using the word metric correctly though: Is that OK? Does anyone has a different way of proving it?
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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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Quasi-Group represented by a graph which is not a Triangle-Free Graph locally

Can each of all quasi-groups be represented by a graph (latin square graph), which is not locally triangle free graph ? Quasi Group can be represented by Latin Square matrix, thus by a Latin ...
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In a loop, if $(xy)x^r = y$ then $x(yx^r)=y$

Consider a loop $L$, that is, a quasigroup with an identity, and recall that a quasigroup $L$ is a set together with a binary operation such that, for every $a$ and $b$ in $L$, the equations $ax=b$ ...
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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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Does every non-empty quasigroup have a left or right identity?

I know that some quasigroups are not loops, meaning they don't have a two-sided identity. But are there non-empty quasigroups that don't even have one-sided identities?
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Non-associative quasigroups [closed]

I want to find more examples of "natural" non-associative quasigroups from any branches of mathematics. For example, $Z_n$ is associative quasigroup. Quasigroup with table 0 1 2 3 4 1 0 3 4 2 ...
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A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for all ...
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A familiar quasigroup - about independent axioms

A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations $a/x=b$ and $y/a=b$. Now I want to extract a ...
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Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
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What is the meaning of division for quasigroups in here?

I am reading this. It says that a quasigroup is a magma in which division is always possible. I'm a little confuse with the meaning of division. Does division means that the inverse operation of $\...
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The unique loop (quasigroup with unit) $L$ of order $5$ satisfying $x^2 = 1$ for all $x \in L$

Recall that a quasigroup is a pair $(Q, \ast)$, where $Q$ is a set and $\ast$ is a binary product $$\ast: Q \times Q \to Q$$ satisfying the Latin square property, namely that for all $x, y \in Q$ ...
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What are the commutative quasigroups satisfying $a/b=b/a$?

There's a harder question lurking behind this question that was just asked. The context is quasigroup theory. A commutative quasigroup can be defined as a set $Q$ together with commutative binary ...
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Identifying all the quasigroups of order $3$ up to isomorphism

According to this, there are $5$ non-isomorphic quasigroups of order $3$. I have been able to find $4$ of them: the cyclic group of order $3$ a commutative quasigroup with $3$ idempotent elements a ...
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Examples of quasigroups with no identity elements

If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general). What would be some examples of quasigroups without an identity element?...
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The smallest quasigroup which is not a group

I'm wondering, which is the smallest quasigroup which is not a group? And how to check it?
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Good introductory book for self-studying quasigroups?

I'm looking for an undergraduate or beginning graduate level text from which to self-learn quasigroup theory. An emphasis on using quasigroups to understand the structure of groups would be ...
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Is there a standard category-theoretic way to express a loop or quasigroup?

The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation. ...