Right group explained

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 while groups can have only one identity and any element can have only one inverse, right groups allow for multiple identity elements and multiple inverse elements.

It can be proven (theorem 1.27 in) that a right group is isomorphic to the direct product of a right zero semigroup and a group, while a right abelian group is the direct product of a right zero semigroup and an abelian group. Left group and left abelian group are defined in analogous way, by substituting right for left in the definitions. The rest of this article 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

with a binary operation ⋅, satisfying the following axioms:

Closure

For all

and

, there is an element c in

such that

c=a ⋅ b

Associativity

For all

a,b,c

(a ⋅ b) ⋅ c=a ⋅ (b ⋅ c)

Left identity element

There is at least one left identity in

. That is, there exists an element

such that

e ⋅ a=a

for all

. Such an element does not need to be unique.

Right inverse elements

For every

and every identity element

, also in

, there is at least one element

, such that

a ⋅ b=e

. Such element

is said to be the right inverse of

with respect to

Examples

Direct product of finite sets

The following example is provided by. 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

and

is simply the cyclic group of order 3, with

as its identity, and

and

as the inverses of each other.

G

table!!e!a!b
e	e	a	b
a	a	b	e
b	b	e	a

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

x ⋅ y=y

, for any

and

Z

table!!1!2
1	1	2
2	1	2

R_gz=G x Z

of these two structures is defined as follows:

The elements of

R_gz

are ordered pairs

(g,z)

such that

is in

and

is in

R_gz

operation is defined element-wise:

Formula 1:

(x,y) ⋅ (u,v)=(xu,v)

The elements of

R_gz

will look like

(e,1),(e,2),(a,1)

and so on. For brevity, let's rename these as

e_1,e_2,a₁

, and so on. The Cayley table of

R_gz

is as follows:

R_gz

table!!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_gz

has two left identities:

e₁

and

e₂

. Some examples:

e2 ⋅ b1=b1

e1 ⋅ a2=a2

Each element has two right inverses. For example, the right inverses of

a₂

with regards to

e₁

and

e₂

are

b₁

and

b₂

, respectively.

a2 ⋅ b1=e1

a2 ⋅ b2=e2

Complex numbers in polar coordinates

Clifford gives a second example involving complex numbers. Given two non-zero complex numbers a and b, the following operation forms a right group:

a ⋅ b=|a|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

a=Aeⁱ

and

b=Beⁱ

, where A and B are the magnitudes and

\alpha

and

\beta

are the arguments (angles) of a and b, respectively.

a ⋅ b

(this is not the regular multiplication of complex numbers) then becomes

Aeⁱ ⋅ Beⁱ=ABeⁱ

. If we represent the magnitudes and arguments as ordered pairs, we can write this as:

Formula 2:

(A,\alpha) ⋅ (B,\beta)=(AB,\beta)

This right group is the direct product of a group (positive 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:

(a+bi) ⋅ (c+di)=a+c+di

, or equivalently:

Formula 3:

(a,b) ⋅ (c,d)=(a+c,d)

A practical example from computer science

Consider the following example from computer science, where a set would be implemented as a programming language type.

be the set of date times in an arbitrary programming language.

be the set of transformations equivalent to adding a duration to an element of

be the set of time zone transformations on elements of

Both

and

are subsets of

T_x

, the full transformation semigroup on

behaves like a group, where there is a zero duration and every duration has an inverse duration. If we treat these transformations as right semigroup actions,

behaves like a right zero semigroup, such that a time zone transformation always cancels any previous time zone transformation on a given date time.

Given any two arbitrary date times

and

(ignore issues regarding representation boundaries), one can find a pair of a duration and a time zone that will transform

into

. This composite transformation of time zone conversion and duration adding is isomorphic to the right group

D x Z

Taking the java.time package as an example,^[5] the sets

X,D

and

would correspond to the class ZonedDateTime, the function plus and the function withZoneSameInstant, respectively. More concretely, for any ZonedDateTime t1 and t2, there is a Duration d and a ZoneId z, such that:

t2 = t1.plus(d).withZoneSomeInstant(z)

The expression above can be written more concisely using right action notation borrowed from group theory as:

t2=t1.d.z

It can also be verified that durations and time zones, when viewed as transformations on date/times, in addition to obeying the axioms of groups and right zero semigroups, respectively, they commute with each other. That is, for any date/time t, any duration d and any timezone z:

t.d.z=t.z.d

This is the same as saying:

d ⋅ z=z ⋅ d

Notes and References

Book: Nagy, Attila. Special classes of semigroups. 2001. Kluwer Academic Publishers. 0-7923-6890-8. Dordrecht. 46240335.
Book: Clifford, A. H.. The algebraic theory of semigroups. Preston, G. B.. 29 June 2014. 978-1-4704-1234-0. Reprinted with corrections. Providence, Rhode Island. 882503487.
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. en. 71. 5. 409–481. 10.1007/s00407-017-0193-8. 1432-0657. 5573778. 28912607.
Clifford. A. H.. 1933. A System Arising from a Weakened Set of Group Postulates. Annals of Mathematics. 34. 4. 865–871. 10.2307/1968703. 1968703. 0003-486X.
Web site: java.time (Java Platform SE 8). 2021-06-03. docs.oracle.com.