Variety of groups with operators
Let be a monoid. The variety of -groups, or variety of groups with operator monoid , is defined as follows. An algebra in this variety is a group , along with a specified action of on by endomorphisms. More precisely, an algebra in this variety has the following operator domain:
- A binary operation for the group multiplication, a unary operation for the group inverse map, and a constant operation for the identity element of the group. These three operations satisfy the universal identities needed to form a group.
- For each element of , a unary operation, subject to three kinds of universal identities: two, to ensure that the action is monoidal, and one, to ensure that each element acts by endomorphisms.
Thus, an algebra of this variety is a group , with a binary operation , a unary operation , a constant operation , and unary operations for every , such that the following are satisfied:
- Identity element:
- Inverse element:
- Endomorphism property for each :
- Composition property for each :
- Identity property (the subscript denotes the identity element of ):
In the special case where is a group, the action of on is by automorphisms.