Right group

A right group, originally called multiple group,{{Cite journal|last=Hollings|first=Christopher D.|date=2017-09-01|title='Nobody could possibly misunderstand what a group is': a study in early twentieth-century group axiomatics|url= |journal=Archive for History of Exact Sciences|language=en|volume=71|issue=5|pages=409–481|doi=10.1007/s00407-017-0193-8|issn=1432-0657|pmc=5573778|pmid=28912607}}{{Cite journal|last=Clifford|first=A. H.|date=1933|title=A System Arising from a Weakened Set of Group Postulates|url=https://www.jstor.org/stable/1968703|journal=Annals of Mathematics|volume=34|issue=4|pages=865–871|doi=10.2307/1968703|jstor=1968703|issn=0003-486X}} is a set $R$ with a binary operation ⋅, satisfying the following axioms:

; Closure

: For all $a$ and $b$ in $R$, there is an element c in $R$ such that $c\; =\; a\; \backslash cdot\; b$.

; Associativity

: For all $a,\; b,\; c$ in $R$, $(a\; \backslash cdot\; b)\; \backslash cdot\; c\; =\; a\; \backslash cdot\; (b\; \backslash cdot\; c)$.

; Left identity element

: There is at least one left identity in $R$. That is, there exists an element $e$ such that $e\; \backslash cdot\; a\; =\; a$ for all $a$ in $R$. Such an element does not need to be unique.

; Right inverse elements

: For every $a$ in $R$ and every identity element $e$, also in $R$, there is at least one element $b$ in $R$, such that $a\; \backslash cdot\; b\; =\; e$. Such element $b$ is said to be the right inverse of $a$ with respect to $e$.

Clifford provides the following example. Take the group $G\; =\; \backslash \{\; e,\; a,\; b\; \backslash \}$, the right zero semigroup $Z\; =\; \backslash \{\; 1,\; 2\; \backslash \}$ 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.

:

class="wikitable" |+$G$ table ! !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 $x\; \backslash cdot\; y\; =\; y$, for any $x$ and $y$ in $Z$.

:

class="wikitable" |+$Z$ table ! !1 !2
1 |1 |2
2 |1 |2

The direct product $R\_\{gz\}\; =\; G\; \backslash times\; Z$ of these two structures is defined as follows:

• The elements of $R\_\{gz\}$ are 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: $(x,\; y)\; \backslash cdot\; (u,\; v)\; =\; (xu,\; v)$

The elements of $R\_\{gz\}$ will look like $(e,\; 1),\; (e,\; 2),\; (a,\; 1)$ and so on. For brevity, let's rename these as $e\_1,\; e\_2,\; a\_1$, and so on. The Cayley table of $R\_\{gz\}$ is as follows:

class="wikitable" |+$R\_\{gz\}$ 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 $R\_\{gz\}$:

• $R\_\{gz\}$ has two left identities: $e\_1$ and $e\_2$. Some examples:
• : $e2\; \backslash cdot\; b1\; =\; b1$
• : $e1\; \backslash cdot\; a2\; =\; a2$
• Each element has two right inverses. For example, the right inverses of $a\_2$ with regards to $e\_1$ and $e\_2$ are $b\_1$ and $b\_2$, respectively.
• : $a2\; \backslash cdot\; b1\; =\; e1$
• : $a2\; \backslash cdot\; b2\; =\; e2$

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

: $a\; \backslash cdot\; b\; =\; |a|\backslash ,\; b$

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 $a\; =\; A\; e^\{i\; \backslash alpha\}$ and $b\; =\; B\; e^\{i\; \backslash beta\}$, where A and B are the magnitudes and $\backslash alpha$ and $\backslash beta$ are the arguments (angles) of a and b, respectively. $a\; \backslash cdot\; b$ (this is not the regular multiplication of complex numbers) then becomes $A\; e^\{i\; \backslash alpha\}\; \backslash cdot\; B\; e^\{i\; \backslash beta\}\; =\; AB\; e^\{i\; \backslash beta\}$. If we represent the magnitudes and arguments as ordered pairs, we can write this as:

: Formula 2: $(A,\; \backslash alpha)\; \backslash cdot\; (B,\backslash beta)\; =\; (AB,\; \backslash beta)$

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.

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:

: $(a\; +\; bi)\; \backslash cdot\; (c\; +\; di)\; =\; a\; +\; c\; +\; di$, or equivalently:

: Formula 3: $(a,b)\; \backslash cdot\; (c,d)\; =\; (a+c,d)$

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

• Let $X$ be the set of date times in an arbitrary programming language.
• Let $D$ be the set of transformations equivalent to adding a duration to an element of $X$.
• Let $Z$ be the set of time zone transformations on elements of $X$.

Both $D$ and $Z$ are subsets of $T\_x$, the full transformation semigroup on $X$. $D$ 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, $Z$ 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 $a$ and $b$ (ignore issues regarding representation boundaries), one can find a pair of a duration and a time zone that will transform $a$ into $b$. This composite transformation of time zone conversion and duration adding is isomorphic to the right group $D\; \backslash times\; Z$.

Taking the java.time package as an example,{{Cite web|title=java.time (Java Platform SE 8 )|url=https://docs.oracle.com/javase/8/docs/api/java/time/package-summary.html|access-date=2021-06-03|website=docs.oracle.com}} the sets $X,\; D$ and $Z$ 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:

: $d\; \backslash cdot\; z\; =\; z\; \backslash cdot\; d$

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Category:Algebraic structures

Category:Semigroup theory