Variety of Lie rings

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Definition

As a plain variety

The variety of Lie rings is the variety of algebras with the operator domain consisting of:

  • A binary operation +
  • A unary operation - (prefix symbol)
  • A constant 0
  • A binary operation [ \ , \ ]

such that the following universal identities are satisfied:

  1. (x + y) + z = x + (y + z)
  2. x + 0 = 0 + x
  3. x + (-x) = (-x) + x = 0
  4. x + y = y + x
  5. [x,y+z] = [x,y] + [x,z]
  6. [x + y,z] = [x,z] + [y,z]
  7. [x,x] = 0
  8. [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0

The identities (1)-(4) say that we get an Abelian group under +, the identities (5) and (6) say the Lie bracket is bilinear, the identity (7) says it is alternating, and the identity (8) is the Jacobi identity.

Properties

A complete listing of the universal algebra-theoretic properties of Lie rings is available at:

Category:Property satisfactions for the variety of Lie rings

Good congruence-related properties