Variety of groups with operators

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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.