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.