Definitions for the Study of the Algebraic Structures over the Quaternioid $\mathbb{R}$ Space

Introduction

The three companion structures of The Algebraic Structures over the Quaternioid $\mathbb{R}$ Space are read through a vocabulary of elements and of identities which the blocks that follow use without restating it. This article fixes that vocabulary once: the element vocabulary — the square, the powers, the unit and the units, the inverse, the radical, the zero divisor, the idempotent, the nilpotent and the square root of a central value — is stated for an arbitrary real-bilinear product and then read on the three; the identity vocabulary — alternating, flexible, Jordan, Jacobi, power-associative and the degree-three identities — is stated with its mutual implications; and the criterion for an empty class is given, the rule by which a class defined by an equation is recognized as empty from the laws of the product alone.

Nothing here is a theorem about the quaternion space; the article is the common grammar of the blocks that follow, and every statement in it is a definition or an implication among definitions. The products and the three structures are those of The 6 Products of the Quaternioid Real Space and The Algebraic Structures over the Quaternioid $\mathbb{R}$ Space, the two articles preceding this one. The algebra product $\tilde q_1\tilde q_2$, which makes the space a division algebra, belongs to the Quaternions category.

The Element Vocabulary

Let $f$ be a real-bilinear rule on $\mathbb{H}$.

Square and powers. The square of $x$ is $x^2=f(x,x)$ and the powers are $x^1=x$, $x^{k+1}=f(x,x^k)$. The square is the only power used in the blocks; the powers $x^k$ for $k\geq3$ are well defined for every rule, being left-nested products of $k$ copies of $x$.

Unit. An element $e$ is a left unit when $f(e,x)=x$ for every $x$, a right unit when $f(x,e)=x$ for every $x$, and a two-sided unit or unit when both. A rule has at most one two-sided unit.

Inverse and units. When the rule has a two-sided unit $e$, the inverse of $x$ is the element $y$ with $f(x,y)=e=f(y,x)$, and $x$ is invertible or a unit when it has an inverse. Without a two-sided unit the word inverse is not used; the blocks say instead that an element is regular for a one-sided unit.

Radical. The radical of the rule is

$$ \operatorname{rad}(f)=\{x:f(x,y)=0=f(y,x)\ \text{for every}\ y\} , $$

the two-sided annihilator of the whole space; it is a subspace, and the rule is nondegenerate when the radical is $\{0\}$.

Zero divisor. A nonzero element $x$ is a left zero divisor when $f(x,y)=0$ for some nonzero $y$, a right zero divisor when $f(y,x)=0$ for some nonzero $y$, and a zero divisor when either. A rule with no zero divisor and a two-sided unit is a division rule, and an algebra whose rule is a division rule is a division algebra.

Idempotent. An idempotent is an element with $x^2=x$; the idempotents form a set containing $0$ and every unit.

Nilpotent. A nilpotent is a nonzero element with $x^k=0$ for some $k$; the case $k=2$, $x^2=0$, is a square-zero element and the case $k=3$, $f(x,f(x,x))=0$, a cube-zero element. Nilpotents are the nonzero elements of the class the blocks call the nilpotent class.

Square root of a central value. The centre of the rule is $\{z:f(z,x)=f(x,z)\ \text{for every}\ x\}$; a square root of a central value $c$ is an element $x$ with $x^2=c$ and $c$ in the centre. A central value may have no square root, one, or a class of them, and the class is the set $\{x:x^2=c\}$.

The Identity Vocabulary

Alternating. The rule is alternating when $f(x,x)=0$ for every $x$.

Flexible. The rule is flexible when $f(f(x,y),x)=f(x,f(y,x))$ for all $x,y$.

Jordan. The rule satisfies the Jordan identity when $f(f(f(x,x),y),x)=f(f(x,x),f(y,x))$ for all $x,y$, that is $(x^2y)x=x^2(yx)$.

Jacobi. The rule satisfies the Jacobi identity when

$$ f(x,f(y,z))+f(y,f(z,x))+f(z,f(x,y))=0 $$

for all $x,y,z$.

Power-associative. The rule is power-associative when $f(f(x,x),x)=f(x,f(x,x))$ for every $x$, so that the two nestings of the cube agree and all powers are unambiguous.

Degree-three identities. The identities of degree three are the identities in which one factor of a product is itself a product: associativity, $f(f(x,y),z)=f(x,f(y,z))$; the left alternative identity, $f(f(x,x),y)=f(x,f(x,y))$; the right alternative identity, $f(f(y,x),x)=f(y,f(x,x))$; and flexibility, which is obtained from associativity by replacing $z$ with $x$ in both sides. A rule is alternative when both alternative identities hold, and associative when so does associativity.

Implications. The following implications among the identities are the ones the blocks use; each is immediate from the definitions and the bilinearity. - $$ \boxed{\;\text{associative}\ \Rightarrow\ \text{flexible, power-associative, Jordan}\;} $$ An associative rule inserts and removes parentheses freely, so the two nestings of the cube agree, the flexibility is two writings of the same triple product, and the Jordan identity is the reading of $x^2yx$ on both sides. - Alternating and bilinear $\Rightarrow$ flexible, power-associative, cube-zero. A bilinear alternating rule is antisymmetric, $f(y,x)=-f(x,y)$, since $f(x+y,x+y)=0$; flexibility is then $f(f(x,y),x)=-f(x,f(x,y))=f(x,f(y,x))$, the power identity is read on the square $f(x,x)=0$, and the cube $f(x,f(x,x))=f(x,0)=0$. - Symmetric and bilinear $\Rightarrow$ flexible, power-associative. A symmetric rule interchanges the two slots everywhere, and the flexibility and the power identity each compare two readings of the same arguments, so the interchange makes them equal. - Alternating and Jacobi $\Rightarrow$ Lie. An alternating bilinear rule satisfying the Jacobi identity is a Lie bracket; that is the definition of the Lie structure.

The Reading on the Three

The table records, for each identity, the structures that satisfy it, on the basis pairs and triples of The Algebraic Structures over the Quaternioid $\mathbb{R}$ Space.

identity	$\mathrm{SPA}$	$\mathrm{APA}$	$\mathrm{GQA}$
alternating	no	yes	no
flexible	yes	yes	no
power-associative	yes	yes	no
Jordan identity	yes	yes	yes
associative	no	no	no
alternative	no	no	no
Jacobi identity	no	yes	no

The reading is the law table of The Algebraic Structures over the Quaternioid $\mathbb{R}$ Space reordered by identity rather than by structure, and it obeys the implications above: the antisymmetric plain structure is the only alternating one, and it is flexible and power-associative and cube-zero by the second implication; the symmetric plain structure is flexible and power-associative by the third; and the general quaternionic structure inherits no law from its product's halves, its first factor being conjugated, so that it is the only one of the three that is neither alternating, nor flexible, nor power-associative.

The Jordan identity holds for all three, the antisymmetric structure included: the Jordan identity is weak, and the structure called the Jordan layer is $\mathrm{SPA}$ because its product is symmetric and the identity is the strongest law it has, not because it is the only one satisfying the identity.

The Criterion for an Empty Class

A class defined by an equation is recognized as empty from the laws of the product alone, and the criterion is one line.

Criterion. Let a class be defined by an equation $P(x)=0$ for a polynomial $P$ built from the product. The class is empty when the equation contradicts a law of the product, that is when every element satisfying the law and the equation would violate an identity the product satisfies; otherwise the class is tested on the basis.

The applications the blocks use are these. The class of nonzero idempotents is empty for the antisymmetric plain structure, since $x^2=x$ reads $0=x$ there, and it is $\{e_t\}$ for the symmetric plain and the general quaternionic structures, a nonzero idempotent being a unit in the first and, in the second, $x^{\natural}x=\langle x\rangle_{\natural}e_t=x$ forcing $x$ into the centre with $x^2=x$ giving $x=e_t$. The class of nonzero nilpotents is empty for the general quaternionic structure, since $x^2=0$ with $x\neq0$ would make $x$ a zero divisor, and there is none; it is the whole space apart from the origin for the antisymmetric plain structure, every nonzero element being square-zero; and it is read on the basis for the symmetric plain structure. The class of units is empty for the antisymmetric plain and the general quaternionic structures, the first having no unit and the second only a left one, while the symmetric plain structure has the two-sided unit $e_t$; and the class of square roots of the central value $-e_t$ is empty for the two structures whose square is a real-valued expression of one sign, $\mathrm{GQA}$ having $x^2=\langle x\rangle_{\natural}e_t$ with $\langle x\rangle_{\natural}\geq0$ and the antisymmetric plain structure having $x^2=0$, while it is nonempty for the symmetric plain structure, the unit vector $\vec e_x$ having the square $-e_t$ there. The algebraic norm functions behind these classes are the two algebraic norms of The 2 Algebraic Norms over the Quaternioid $\mathbb{R}$ Space, below this article in this category, whose level sets are read there.

Summary

The element vocabulary of the three companion structures is the square and the powers, the unit and the units, the inverse, the radical, the zero divisor, the idempotent, the nilpotent and the square root of a central value, each defined for a general real-bilinear product and read on the three. The identity vocabulary is alternation, flexibility, the Jordan identity, the Jacobi identity, power-associativity and the degree-three identities, with the implications: associative implies flexible, power-associative and Jordan; alternating with bilinearity implies antisymmetry, flexibility, power-associativity and cube-zero; symmetric with bilinearity implies flexibility and power-associativity; and alternating with Jacobi is the Lie structure.

The reading on the three gives the antisymmetric plain structure as the only alternating one and the only one satisfying Jacobi, and as flexible and power-associative; the symmetric plain structure as flexible and power-associative, with the unit $e_t$; none of the three as associative; the Jordan identity as holding for all three; and the general quaternionic structure as inheriting no law from its product's halves. A class is empty exactly when its defining equation contradicts a law of the product, and the criterion is applied to the classes of idempotents, nilpotents, units and square roots of a central value.

Summary of Notation

Symbol	Meaning
$x^2=f(x,x)$, $x^{k+1}=f(x,x^k)$	square and powers
left, right, two-sided unit $e$	$f(e,x)=x$, $f(x,e)=x$, both
inverse of $x$	$y$ with $f(x,y)=e=f(y,x)$
$\operatorname{rad}(f)=\{x:f(x,y)=0=f(y,x)\ \forall y\}$	the radical
zero divisor	nonzero $x$ with $f(x,y)=0$ or $f(y,x)=0$, $y\neq0$
idempotent	$x^2=x$
nilpotent, square-zero, cube-zero	$x^k=0$; $x^2=0$; $f(x,f(x,x))=0$
square root of a central value	$x$ with $x^2=c$, $c$ central
alternating, flexible	$f(x,x)=0$; $f(f(x,y),x)=f(x,f(y,x))$
Jordan, Jacobi	$(x^2y)x=x^2(yx)$; $f(x,f(y,z))+f(y,f(z,x))+f(z,f(x,y))=0$
power-associative, degree three	$f(f(x,x),x)=f(x,f(x,x))$; identities with a nested factor