In mathematics, a right group 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 one-sided identity elements and multiple one-sided 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, is a set R {\displaystyle R} with a binary operation ⋅, satisfying the following axioms:
Closure For all a {\displaystyle a} and b {\displaystyle b} in R {\displaystyle R} , there is an element c in R {\displaystyle R} such that c = a ⋅ b {\displaystyle c=a\cdot b} . Associativity For all a , b , c {\displaystyle a,b,c} in R {\displaystyle R} , ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} . Left identity element There is at least one left identity in R {\displaystyle R} . That is, there exists an element e {\displaystyle e} such that e ⋅ a = a {\displaystyle e\cdot a=a} for all a {\displaystyle a} in R {\displaystyle R} . Such an element does not need to be unique. Right inverse elements For every a {\displaystyle a} in R {\displaystyle R} and every identity element e {\displaystyle e} , also in R {\displaystyle R} , there is at least one element b {\displaystyle b} in R {\displaystyle R} , such that a ⋅ b = e {\displaystyle a\cdot b=e} . Such element b {\displaystyle b} is said to be the right inverse of a {\displaystyle a} with respect to e {\displaystyle e} .
Examples
Direct product of finite sets Clifford provides the following example. Take the group G = { e , a , b } {\displaystyle G=\{e,a,b\}} , the right zero semigroup Z = { 1 , 2 } {\displaystyle Z=\{1,2\}} and construct a right group R g z {\displaystyle R_{gz}} as the direct product of G {\displaystyle G} and Z {\displaystyle Z} .
G {\displaystyle G} is simply the cyclic group of order 3, with e {\displaystyle e} as its identity, and a {\displaystyle a} and b {\displaystyle b} as the inverses of each other.
Z {\displaystyle Z} is the right zero semigroup of order 2. Notice the each element repeats along its column, since by definition x ⋅ y = y {\displaystyle x\cdot y=y} , for any x {\displaystyle x} and y {\displaystyle y} in Z {\displaystyle Z} .
The direct product R g z = G × Z {\displaystyle R_{gz}=G\times Z} of these two structures is defined as follows:
The elements of R g z {\displaystyle R_{gz}} are ordered pairs ( g , z ) {\displaystyle (g,z)} such that g {\displaystyle g} is in G {\displaystyle G} and z {\displaystyle z} is in Z {\displaystyle Z} . The R g z {\displaystyle R_{gz}} operation is defined element-wise: Formula 1: ( x , y ) ⋅ ( u , v ) = ( x u , v ) {\displaystyle (x,y)\cdot (u,v)=(xu,v)}
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