Left quasifield

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Definition

A left quasifield is a set Q equipped with:

  • A (infix) binary operation +, called the addition or additive operation.
  • A unary operation -, called the additive inverse.
  • A constant 0, called zero.
  • A (infix) binary operation *, called the multiplication.
  • A constant 1 such that 1 \ne 0.

Such that:

  • (Q,+,0,-) is a group.
  • (Q \setminus \{ 0 \},*,1,) is a Moufang loop with multiplication * and identity element 1.
  • \! a * (b + c) = (a * b) + (a * c) (i.e., we have the left distributive law).
  • \! a * x = (b * x) + c has exactly one solution in x for any fixed a,b,c \in Q with a \ne b.