Intereting Posts

Is characterisation of degree 2 nilpotent matrices (i.e. $M^2=0$) known?
The narrow class group of an order of a quadratic field and the genera of binary quadratic forms
What is the name of the logical puzzle, where one always lies and another always tells the truth?
Is $\mathbb Z _p^*=\{ 1, 2, 3, … , p-1 \}$ a cyclic group?
Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$
Entropy of sum of random variables
Mean of gamma distribution.
counting full bipartite matchings
Exact sequence arising from symplectic manifold
Should I write out stuff?
Uniqueness of prime ideals of $\mathbb F_p/(x^2)$
What's behind the Banach-Tarski paradox?
What comes after exponents?
Proving the inverse of a matrix equals $I_n-\frac{1}{n-1}A$
horizontal vector in tangent bundle

What’s the difference of a monoid and a group? I’m reading this book and it says that a group is a monoid with invertibility and this property is made to solve the equation $x \ast m=e$ and $m \ast x=e$ for $x$, where $m$ is any element of the structure.

I got confused because it’s similar to the monoid’s commutativity property which says that $m \ast n=n*m$ for all $m, n \in M$.

- Holomorph is isomorphic to normalizer of subgroup of symmetric group?
- Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.
- Find the order of a subgroup of $S_5$ generated by two elements
- Character on conjugacy classes
- Single $\text{GL}_n(\mathbb{C})$-conjugacy class, dimension as algebraic variety?
- Why is $xyzx^{-1}y^{-1}z^{-1}$ a commutator?

- List all subgroups of the symmetry group of $n$-gon
- Normal subgroups of free groups: finitely generated $\implies$ finite index.
- Is there a non-cyclic group with every subgroup characteristic?
- Normal subgroup if conjugate subgroup is subset
- If $G$ is isomorphic to all non-trivial cyclic subgroups, prove that $G\cong \mathbb{Z}$ or $G\cong \mathbb{Z}_p$
- An exact sequence of unit groups
- {5,15,25,35} is a group under multiplication mod 40
- Sylow 2-subgroup of GL(2,R)
- What is this automorphism-related subgroup?
- Subgroups of a direct product

Your confusion arises from the fact that you are using the same letter in both equations. It would be better to say that invertibility is the property that for every $m$ there is a solution to the equation $m*x = e$, and a solution to the equation $y*m=e$. You can then *prove* that the solutions will in fact be the same, since

$$y = y*e = y*(m*x) = (y*m)*x = e*x = x.$$

Moreover, while it is true that the two equations together imply that $mx=xm$, this is not equivalent to commutativity. To be clear, commutativity would be

For all $a$ and all $b$, $ab=ba$.

Here you have only

If $x$ is the solution to $mx=e$, then $mx=xm$.

That is, you are only guaranteed that a *particular* element commutes with each $m$, not that *every* element commutes with every element.

Consider the usual “axioms” of a group. the ingredients are a set $S$, and a function $\cdot\colon S\times S\to S$, which we write using infix notation (so we write $a\cdot b$ or $ab$ instead of $\cdot(a,b)$). Then we require:

**Associativity.**$\cdot$ is associative: $a(bc) = (ab)c$ for all $a,b,c\in S$.**Existence of neutral element.**There exists an element $e\in S$ such that for all $a\in S$, $ae=ea=a$.**Existence of inverses.**For each $a\in S$ there exists $b\in S$ such that $ab=ba=e$, where $e$ is a neutral element as in 2.

If we relax the requirements that all three conditions get satisfied, we get more general structures (but the more general the structure, the less we can say about them).

- If you drop all three conditions, you get a
**magma**. - If you drop the second and third condition but keep the first, requiring only that the operation be associative, you get a
**semigroup.** - If you drop the third condition but keep the first and second, requiring that the operation be associative and that there be a neutral element, you get a
**monoid.** - If you keep all three conditions, you get a
**group.**

There are other things you can do; it does not make sense to drop the second and keep the third condition.

If you drop the first (associativity), then can relax the conditions a bit and ask that all equations of the form $ax=b$ and $ya=b$ have solutions, but not requiring that the operation be associative. That gives you a **quasigroup.** If you require that all such equations have solutions **and** that there be an identity, you get a **loop**. This is equivalent to asking that conditions 2 and 3 be satisfied, but not condition 1.

Within each category you can put other conditions. There are “cancellation semigroups”, which are semigroups in which $ax=ay$ implies $x=y$. There are “inverse semigroups” which, perhaps confusingly, does *not* mean that condition 3 is satisfied (makes no sense if we don’t have condition 2), but rather that for every $a$ there exists a $b$ such that $aba=a$ and $bab=b$. And so on and so forth. Lots of different wrinkles to be seen in there.

The difference is that an element of a monoid doesn’t have to have inverse, while an element of a group does. For example, $\mathbb N$ is a monoid under addition (with identity $0$) but not a group, since for any $n,m\in \mathbb N$ if $n$ or $m$ is not $0$ then $n+m\neq 0$.

Elements of a monoid do not *necessarily* have inverse elements, while those of a group do. See

http://en.wikipedia.org/wiki/Monoid

There are 4 axioms that define a group, one of which is the presence of inverse elements. Monoids only need to satisfy the other 3.

First, not every monoid has the commutative property. A monoid which is commutative is called a commutative monoid.

Now, to answer your question. Not every element in a monoid has the inverse element. However, if an element $m$ in a commutative monoid has a left inverse, i.e. $x * m = e$, then $x$ is the inverse of $m$ because $m * x = e$ by the commutative property – you can only get this in a commutative monoid.

The difference between a monoid and a group is what you said, a group is a monoid with the invertibility property.

Edit:

In response to the OP’s later comment – he saw a sentence “A group is commutative, or abelian, if it is so as a monoid.” in the book he cited. It means that a group is also a monoid and if it is commutative when viewing it as a monoid, then it is a commutative(abelian) group.

- Let $K$ be a Sylow subgroup of a finite group $G$. Prove that if $x \in N(K)$ and the order of $x$ is a power of $p$, then $ x \in K$.
- Quotient of the ring of integers of a number field by a prime ideal
- Complex number: calculate $(1 + i)^n$.
- References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$
- When does Schwarz inequality become an equality?
- Gaussian proof for the sum of squares?
- Creating a special vector from two vectors
- counting the number of elements in a conjugacy class of $S_n$
- $(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K$
- Prove that the number of jump discontinuities is countable for any function
- How to determine in polynomial time if a number is a product of two consecutive primes?
- Given a, b How many solutions exists for x, such that: $a \bmod{x}=b $
- Drawing sine and cosine waves
- Counterexamples for “every linear map on an infinite dimensional complex vector space has an eigenvalue”
- Prove that the convex hull of a set is the smallest convex set containing that set