PhyslibAlpha.ProbabilisticTheory.Algebra.Alternative

Every semigroup $A$ is an alternative algebra. That is, if the multiplication in $A$ is associative, then the associator $(x, y, z) = (xy)z - x(yz)$ satisfies the alternating property.

For elements $x, y, z$ in an algebra $A$, the associator is defined as the ternary operation $(x, y, z) = (xy)z - x(yz)$, which measures the failure of the associative law.

In an alternative algebra $A$, for any elements $x, y, z, w \in A$, the associator satisfies the left-additive identity $(x + y, z, w) = (x, z, w) + (y, z, w)$, where the associator is defined as $(x, y, z) = (xy)z - x(yz)$.

In an algebra $A$, the associator of three elements is defined as $(x, y, z) = (xy)z - x(yz)$. For any elements $x, y, z, w \in A$, the associator is additive in its middle argument: $$(x, y + z, w) = (x, y, w) + (x, z, w)$$

For any elements $x, y, z$, and $w$ in an algebra $A$, the associator $(x, y, z) = (xy)z - x(yz)$ is additive in its third argument, satisfying the identity $(x, y, z + w) = (x, y, z) + (x, y, w)$.

For any elements $x, y, z, w$ in an algebra $A$, let the associator be defined as $(x, y, z) = (xy)z - x(yz)$. Then the following identity holds: $$(xy, z, w) - (x, yz, w) + (x, y, zw) = (x, y, z)w + x(y, z, w)$$ This is known as Teichmüller's identity and holds in any non-associative algebra.

In an alternative algebra $A$, for any elements $x, y \in A$, the associator $(x, x, y)$ is equal to $0$.

In an alternative algebra $A$, for any elements $x, y \in A$, the associator of $y, x, x$ is zero, which is expressed as $(y, x, x) = 0$. Using the definition of the associator, this implies the right alternative law $(yx)x = y(xx)$.

In an alternative algebra $A$, for any elements $x, y, z \in A$, the associator $[x, y, z] = (xy)z - x(yz)$ is skew-symmetric with respect to the first two arguments, satisfying $[x, y, z] = -[y, x, z]$.

In an alternative algebra $A$, for any elements $x, y, z \in A$, the associator $(x, y, z) = (xy)z - x(yz)$ satisfies the identity $(x, y, z) = -(x, z, y)$.

In an alternative algebra $A$, the associator $(x, y, z) = (xy)z - x(yz)$ is cyclically invariant. That is, for any elements $x, y, z \in A$, it holds that $(x, y, z) = (y, z, x)$.

In an alternative algebra $A$, for any elements $x, y \in A$, the associator $(x, y, x)$, defined as $(xy)x - x(yx)$, is equal to $0$.

In an alternative algebra $A$, for any two elements $x, y \in A$, the flexible law $x(yx) = (xy)x$ holds.

In an alternative algebra $A$, for any elements $x, y, z \in A$, the left Moufang identity holds: $$(x(zx))y = x(z(xy))$$ where the associator $(a, b, c) = (ab)c - a(bc)$ is alternating.

In an alternative algebra $A$, for any $x, y, z \in A$, the following identity holds: $$(x, y, zx) = x(y, z, x)$$ where $(a, b, c) = (ab)c - a(bc)$ denotes the associator. This result is known as McCrimmon's left bumping formula, a key alternative-algebra identity used in the study of Albert matrices.