Absolute presentation of a group

In mathematics, an absolute presentation is one method of defining a group.
Recall that to define a group formula_1 by means of a presentation, one specifies a set formula_2 of generators so that every element of the group can be written as a product of some of these generators, and a set formula_3 of relations among those generators. In symbols:
Informally formula_1 is the group generated by the set formula_2 such that formula_7 for all formula_8. But here there is a tacit assumption that formula_1 is the "freest" such group as clearly the relations are satisfied in any homomorphic image of formula_1. One way of being able to eliminate this tacit assumption is by specifying that certain words in formula_2 should not be equal to formula_12 That is we specify a set formula_13, called the set of irrelations, such that formula_14 for all formula_15.
Formal definition.
To define an absolute presentation of a group formula_1 one specifies a set formula_2 of generators, a set formula_3 of relations among those generators and a set formula_13 of irrelations among those 
generators. We then say formula_1 has absolute presentation
provided that:
A more algebraic, but equivalent, way of stating condition 2 is:
Remark: The concept of an absolute presentation has been fruitful in fields such as algebraically closed groups and the Grigorchuk topology.
In the literature, in a context where absolute presentations are being discussed, a presentation (in the usual sense of the word) is sometimes referred to as a relative presentation, which is an instance of a retronym.
Example.
The cyclic group of order "8" has the presentation
But, up to isomorphism there are three more groups that "satisfy" the relation formula_33 namely:
However none of these satisfy the irrelation formula_37. So an absolute presentation for the cyclic group of order 8 is:
It is part of the definition of an absolute presentation that the irrelations are not satisfied in any proper homomorphic image of the group. Therefore:
Is "not" an absolute presentation for the cyclic group of order 8 because the irrelation formula_40 is satisfied in the cyclic group of order 4.
Background.
The notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem for algebraically closed groups.
A common strategy for considering whether two groups formula_41 and formula_42 are isomorphic is to consider whether a presentation for one might be transformed into a presentation for the other. However algebraically closed groups are neither finitely generated nor recursively presented and so it is impossible to compare their presentations. Neumann considered the following alternative strategy: 
Suppose we know that a group formula_41 with finite presentation formula_44 can be embedded in the algebraically closed group formula_45 then given another algebraically closed group formula_46, we can ask "Can formula_41 be embedded in formula_46?"
It soon becomes apparent that a presentation for a group does not contain enough information to make this decision for while there may be a homomorphism formula_49, this homomorphism need not be an embedding. What is needed is a specification for formula_45 that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.