Right group

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 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:
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 ) 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 a 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 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, is a set "G" with a binary operation ⋅, satisfying the following axioms:
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. Take the group G = { "e, a, b" }, the right zero semigroup Z = { 1, 2 } and construct a right group Rgz 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. 
Z is the right zero semigroup of order 2. Notice the each element repeats along its column, since by definition y ⋅ v = v.
The direct product Rgz = G x Z of these 2 structures is defined as follows:
The elements of Rgz 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 Rgz, whose Caley table is the following:
Here are some facts about R:
Complex numbers in polar coordinates.
Clifford gives a second example involving complex numbers. Given two complex numbers a and b, the following operation is a right group:
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 formula_3 and formula_4, where A and B are the magnitudes and formula_5 and formula_6 are the arguments (angles) of a and b, respectively. formula_7 (this is not the regular multiplication of complex numbers) then becomes formula_8. If we represent the magnitudes and arguments as ordered pairs, we can write this as:
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: