Left-orderable skew braces

DefinitionLeft-orderable multiplication#✓fully proved

The multiplication of a set \(B\) is left-orderable when there is a strict total order \(<\) on \(B\) that is invariant under left multiplication: for all \(a, b, c \in B\), if \(b < c\) then \(a b < a c\).

def LeftOrderableMul (B : Type*) [Mul B] : Prop :=
  ∃ r : B → B → Prop, IsStrictTotalOrder B r ∧ ∀ a b c : B, r b c → r (a * b) (a * c)

Used by 1

Used byLeft-orderability of the additive group of a skew brace (Kourovka 19.49)

DefinitionLeft-orderable addition#✓fully proved

The addition of a set \(B\) is left-orderable when there is a strict total order \(<\) on \(B\) that is invariant under left addition: for all \(a, b, c \in B\), if \(b < c\) then \(a + b < a + c\).

def LeftOrderableAdd (B : Type*) [Add B] : Prop :=
  ∃ r : B → B → Prop, IsStrictTotalOrder B r ∧ ∀ a b c : B, r b c → r (a + b) (a + c)

Used by 1

Used byLeft-orderability of the additive group of a skew brace (Kourovka 19.49)

DefinitionLeft-orderability of the additive group of a skew brace (Kourovka 19.49)#✓fully proved

The statement that for every skew brace \(B\), that is, a set with two group structures, an addition \(+\) (not necessarily commutative) and a multiplication \(\cdot\), satisfying \(a \cdot (b + c) = a \cdot b - a + a \cdot c\) for all \(a, b, c \in B\): if the multiplication of \(B\) is left-orderable, then the addition of \(B\) is left-orderable.

def skewBraceAdditiveLeftOrderable : Prop :=
  ∀ (B : Type) [AddGroup B] [Group B], (∀ a b c : B, a * (b + c) = a * b - a + a * c) →
    LeftOrderableMul B → LeftOrderableAdd B

Uses 2 · Used by 1

Statement usesLeft-orderable addition · Left-orderable multiplication

Used byA skew brace whose additive group is not left-orderable

TheoremA skew brace whose additive group is not left-orderable#●ready to prove

There is a skew brace \(B\), that is, a set with two group structures, an addition \(+\) and a multiplication \(\cdot\), satisfying \(a \cdot (b + c) = a \cdot b - a + a \cdot c\) for all \(a, b, c \in B\), whose multiplication is left-orderable but whose addition is not left-orderable. That is, the statement of Kourovka Problem 19.49 fails.

Nasybullov's example (Example 3.1) is \(\mathbb{Z}\) with addition \(a \oplus b = a + (-1)^a b\) and multiplication \(a \odot b = a + b\): the multiplicative group is infinite cyclic, hence left-orderable, while the additive group is infinite dihedral and has an element of order \(2\), so it is not left-orderable. The Kourovka Archive answers 19.49 "No, not always".

public theorem_wanted skewBraceAdditiveLeftOrderable_refuted : ¬ skewBraceAdditiveLeftOrderable

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 19.49 (V. Lebed, L. Vendramin); resolution: T. Nasybullov, Connections between properties of the additive and the multiplicative groups of a two-sided skew brace, J. Algebra 540 (2019), 156-167 (arXiv:1809.09418), Example 3.1: on ℤ, a ⊕ b = a + (−1)^a b and a ⊙ b = a + b define a skew brace with infinite cyclic multiplicative group and infinite dihedral additive group, which has 2-torsion and so is not left-orderable, answering Problem 19.49; Kourovka Archive, Problem 19.49: No, not always, https://arxiv.org/abs/1809.09418.

Uses 1

Statement usesLeft-orderability of the additive group of a skew brace (Kourovka 19.49)

3 fully proved1 ready to prove

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