Variety of groups with operators

Definition
Let $$M$$ be a monoid. The variety of $$M$$-groups, or variety of groups with operator monoid $$M$$, is defined as follows. An algebra in this variety is a group $$G$$, along with a specified action of $$M$$ on $$G$$ by endomorphisms. More precisely, an algebra in this variety has the following operator domain:


 * 1) 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.
 * 2) For each element of $$M$$, 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 $$G$$, with a binary operation $$*$$, a unary operation $${}^{-1}$$, a constant operation $$e$$, and unary operations $$\mu_a$$ for every $$a \in M$$, such that the following are satisfied:


 * Associativity: $$(g * h) * k = g * (h * k) \ forall \ g,h,k \in G$$
 * Identity element: $$g * e = e * g = g \ \forall \ g,h,k, \in G$$
 * Inverse element: $$g * g^{-1} = g^{-1} * g = e \ \forall \ g \in G$$
 * Endomorphism property for each $$a \in M$$: $$\mu_a(g * h) = \mu_a(g) * \mu_a(h) \ \forall \ g \in G$$
 * Composition property for each $$a, b \in M$$: $$\mu_{ab}(g) = \mu_a(\mu_b(g)) \ \forall \ g \in G$$
 * Identity property (the subscript $$e$$ denotes the identity element of $$M$$): $$\mu_e(g) = g \ \forall \ g \in G$$

In the special case where $$M$$ is a group, the action of $$M$$ on $$G$$ is by automorphisms.