H-group

This is a variation of group|Find other variations of group | Read a survey article on varying group

Definition

A H-group is a set $G$ equipped with the structure of a topological space and three operations:

Operation name	Arity of operation	Operation description and notation
Multiplication or product	2	A binary operation $*:G\times G\to G$ (infix operator) termed the multiplication or product. The product of $x$ and $y$ is denoted $x*y$.
Identity element (or neutral element)	0	A 0-ary operation which gives a constant element, denoted by $e$ (sometimes also as $1$), termed the identity element or neutral element.
Inverse map	1	A unary operation $\iota :G\to G$ (superscript operator) termed the inverse map. The inverse of $x$ is denoted $\iota (x)$.

satisfying the following three compatibility conditions:

Condition name	Arity of mappings about which we are making homotopy assertion	Condition description	Comments
Homotopy version of associativity	3	Consider the two mappings $G\times G\times G\to G$ given by $(a,b,c)\mapsto (a*b)*c$ and $(a,b,c)\mapsto a*(b*c)$. These two maps are homotopic to each other as maps from $G\times G\times G\to G$, where the former is equipped with the product topology arising from the topology on $G$.	Note that if the multiplication is associative, then it is homotopy associative, because in that case the two mappings are exactly equal.
Homotopy version of identity element (or neutral element)	1	Consider the following three mappings $G\to G$: the identity map $a\mapsto a$, the map $a\mapsto a*e$, and the map $a\mapsto e*a$. All of these are homotopic to each other.	Note that if $e$ is an identity element in the usual sense of the word, then the three maps are equal and hence homotopic.
Homotopy version of inverse element	1	Consider the map $G\to G$ given by $a\mapsto a*\iota (a)$. This map is homotopic to the constant map $a\mapsto e$.	Note that if $\iota (a)$ is actually a two-sided inverse of $a$ for all $a$, the maps are equal identically.

It's important to note here that unlike the ordinary associativity, identity element, and inverse element conditions, the homotopy versions cannot be checked separately for each tuple of elements. Rather, to check for the truth of the homotopy version, we need to look at the mappings (such as $(a,b,c)\mapsto (a*b)*c$) in their entirety as we try to homotope them.