Right group

In mathematics, a right group [1][2] is an algebraic structure consisting of a set together with a binary operation that combines two elements into a third element while obeying the right group axioms. The right group axioms are similar to the group axioms, but allow for one-sided identity elements and one-sided inverse elements, as opposed to groups where both identities and inverses are two-sided. More precisely, if G is a group, e is the identity of G, a an arbitrary element of G and a' the inverse of a, the following are always true for any group:

• e ⋅ a = a ⋅ e = a
• a' ⋅ a = a ⋅ a' = e

The rules above don't apply to right groups. For a right group, it is possible to have multiple left identities, and for each left identity any element will have a respective right inverse.

It can be proven (theorem 1.27 in [2]) that a right group is isomorphic to the direct product of a right zero semigroup and a group, while a right abelian group [1] is the direct product of a right zero semigroup and a abelian group. Left group [1][2] and left abelian group[1] are defined in analogous way, by substituting right for left in the definitions. The rest of this text will be mostly concerned about right groups, but everything applies to left groups by doing the appropriate right/left substitutions.

Definition

A right group, originally called multiple group,[3][4] is a set G with a binary operation ⋅, satisfying the following axioms:[4]

Closure

For all a and b in G, there is an element c in G, such that c = a ⋅ b.

Associativity

For all a, b, c in G, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Identity elements

For all a in G, there is at least one left identity e, also in G, such that e ⋅ a = a. Such element doesn't need to be unique.

Inverse elements

For every a in G and every identity element e, also in G, there is at least one element b in G, such that a ⋅ b = e. Such element b is said to be the right inverse of a with respect to e.

Examples

Direct product of finite sets

The following example is provided by.[4] Take the group G = { e, a, b }, the right zero semigroup Z = { 1, 2 } and construct a right group R_gz as the direct product of G and Z.

G is simply the cyclic group of order 3, with e as its identity, and a and b as the inverses of each other.

G

	e	a	b
e	e	a	b
a	a	b	e
b	b	e	a

Z is the right zero semigroup of order 2. Notice the each element repeats along its column, since by definition y ⋅ v = v.

Z

	1	2
1	1	2
2	1	2

The direct product R_gz = G x Z of these 2 structures is defined as follows:

• The elements of R_gz are the ordered pairs (g, z), such that g is in G and z is in Z.
• The R_gz operation is defined element-wise:
  • Formula 1: ${\displaystyle (x,y)\cdot (u,v)=(xu,v)}$

The elements of R_gz will look like (e, 1), (e, 2), (a, 1) and so on. For convenience, let's rename these as e₁, e₂, a₁, and so on. This renaming creates another structure, let's say R, isomorphic to R_gz, whose Caley table is the following:

R ≈ G x Z

	e1	a1	b1	e2	a2	b2
e1	e1	a1	b1	e2	a2	b2
a1	a1	b1	e1	a2	b2	e2
b1	b1	e1	a1	b2	e2	a2
e2	e1	a1	b1	e2	a2	b2
a2	a1	b1	e1	a2	b2	e2
b2	b1	e1	a1	b2	e2	a2

Here are some facts about R:

• R has two left identities: e₁ and e₂.
• Each element has 2 right inverses. Example: The right inverse of a₂ with regards to e₁ and e₂ are b₁ and b₂, respectively.

Complex numbers in polar coordinates

Clifford gives a second example[4] involving complex numbers. Given two complex numbers a and b, the following operation is a right group:

• ${\displaystyle a\cdot b=\mid a\mid b}$

All complex numbers with modulus equal to 1 are left identities, and all complex numbers will have a right inverse with respect to any left identity.

The inner structure of this right group becomes clear when we use polar coordinates. Let ${\displaystyle a=Ae^{i\alpha }}$ and ${\displaystyle b=Be^{i\beta }}$, where A and B are the magnitudes and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are the arguments (angles) of a and b, respectively. ${\displaystyle a\cdot b}$ (this is not the regular multiplication of complex numbers) then becomes ${\displaystyle Ae^{i\alpha }\cdot Be^{i\beta }=ABe^{i\beta }}$. If we represent the magnitudes and arguments as ordered pairs, we can write this as:

• Formula 2: ${\displaystyle (A,\alpha )\cdot (B,\beta )=(AB,\beta )}$.

This right group is the direct product of a group (real numbers under multiplication) and a right zero semigroup induced by the real numbers. Structurally, this is identical to formula 1 above. In fact, this is how all right group operations look like when written as ordered pairs of the direct product of their factors.

Complex numbers in cartesian coordinates

If we take the and complex numbers and define an operation similar to example 2 but use cartesian instead of polar coordinates and addition instead of multiplication, we get another right group, with operation defined as follows:

• ${\displaystyle (a+bi)\cdot (c+di)=a+c+di}$, or equivalently:

• Formula 3: ${\displaystyle (a,b)\cdot (c,d)=(a+c,d)}$

References

1. Nagy, Attila (2001). Special classes of semigroups. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-6890-8. OCLC 46240335.
2. Clifford, A. H. The algebraic theory of semigroups. Preston, G. B. (Reprinted with corrections ed.). Providence, Rhode Island. ISBN 978-1-4704-1234-0. OCLC 882503487.
3. Hollings, Christopher D. (2017-09-01). "'Nobody could possibly misunderstand what a group is': a study in early twentieth-century group axiomatics". Archive for History of Exact Sciences. 71 (5): 409–481. doi:10.1007/s00407-017-0193-8. ISSN 1432-0657. PMC 5573778. PMID 28912607.
4. Clifford, A. H. (1933). "A System Arising from a Weakened Set of Group Postulates". Annals of Mathematics. 34 (4): 865–871. doi:10.2307/1968703. ISSN 0003-486X.