# 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.