ALGHCAGloss. Glossary for Loops, Groups, Hoops, Cliffords.

Roger Beresford. 10th. Mar. 2012.

NotebookFileName[](*10/3/12 17.01 version.*)

Introduction,

Negative, real, complex, quaternionic and octonionic numbers, subtraction, and the signs {+, i, -, -i}, are a degenerate subset of mathematics, restricted to even symmetry. Algebraic loops (quasi-groups with 1) define general multiplication for Natural, Rational, and Primal (continuous, absolute, half line) (italic names introduce new mathematical concepts) unsigned numbers and are created without defining division or subtraction. They provide Cayley tables for the multiplication of vecs, sets of {direction, unsigned coefficient} pairs. Signs can only be defined for loops having Cr (the cyclic group with r elements) as central subgroups. Such loops can be folded to smaller signed tables where the r-fold symmetry is represented by s (with ((s )^r)=1) as a generalized sign. This introduces r’th roots of unity as generalised signs.  (r=2 and 4 give real and complex algebras and impose even symmetry). Folding creates algebras with generalised addition/subtraction as a second operation and folds vecs to vectors; Moufang loops (which have a division property ab/b=a=ba/b) fold to vector-division algebras.
Frobenius showed that associative loops (i.e. groups) conserve their determinants (up to a sign) on vector multiplication. This symmetry-conserving property is possessed by all associative (group) and a few non-associative (octonionic 𝕆) Moufang loops, and by algebras folded from conservative loops. Vectors in these algebras have multiplicative inverses that split into partial fractions with determinant factors as numerators. 
I propose the name hoops for these symmetry-conserving partial-fraction-division algebras. I call the factors sizes and the list of conserved sizes the shape of a vector. Noether’s theorem, relating conserved symmetries to particles and forces, implies that hoops describe physical mathematics. Anti-symmetric hoops are the Clifford (Geometric) algebras that generalise many concepts to any number of dimensions. 
Most Hoops have multiple sizes. (Real, complex, quaternion, octonion, Planck and even-order Clifford algebras are degenerate, monosized, hoops.) When two vectors have disparate zero sizes, the product loses any sizes that are only present in one multiplicand - it is projected onto a sub-algebra of reduced symmetry. To maintain the key conservation property, the lost sizes are ejected as (left or right) remainders. Multiplication (and division, which is multiplication by an inverse vector) can create a product and two remainders. This implements vector-division-by-zero. Different algebras (of the same length) conserve different (but overlapping) sets of conserved properties. The analogy with particle physics, where different forces conserve overlapping sets of properties, has yet to be explored.
These concepts were developed empirically, but have (for groups) a theoretical basis in Representation Theory.
This material was originally developed in the ALGHC package [], which includes an empirical database of loops, groups, hoops and Clifford algebras, together with many procedures to create, identify, and apply them. It relied heavily on Cayley tables, and made little reference to standard Group theory. It was limited by the difficulty of finding symbolic sizes and by the lack of a mathematical basis. Work with G. Gerrard led to a reformulation (restricted to groups) in terms of irreducible matrix representations (irreps). These are based in representation theory. They give permutation group descriptions as well as Cayley tables and allow the calculation of vector sizes and inverses. The GroupHoop package [] has a database derived from GAP, and provides simpler access to many small groups and hoops.

Glossary.

a,b,..or (Subscript[a, 1]),(Subscript[a, 2]), Elements representing Primal (unsigned, non-negative) numbers, coefficients in a director.

A	A "Vec", an indexed set {a,b…} of m primal elements; also written (∑_(i=1)^m)(Subscript[a, i])(Subscript[d, i]). where (Subscript[d, i]) is a direction.

A, B, V Vectors, indexed sets {a,b…} of m/r “r-signed” elements; also written (∑_(i=1)^(m/r))(Subscript[a, i])(Subscript[x, i]). (They take on
	 all the properties of vectors in some algebras. In Geometric Algebra Vector is confusing so I use
	 monovector by analogy with bivector etc..
	 
 𝔸=0⋃(SuperPlus[ℝ]) The Primal (unsigned continuous, Absolute) numbers, the union of zero and the half line.

additive elimination. The generalization of subtraction A+(-A)=0 to A+(Subscript[A, i])+(Subscript[A, j])+..=0. (Subscript[A, i]) etc are roots of unity.
          This is the result of folding a loop, which equivalences the sum of a set of roots of unity to zero.
          
Ai= (∑_j)((∑_i)(Subscript[η, i])(∂_(Subscript[a, i ]))(Subscript[f, j ])) /(Subscript[f, j]). The multiplicative inverse of A. Calculated by hoopInverse[A], cInverse[A],
	gapInverse[A]. Ai.A={1,0,..} Also found by adding the inverses of non-zero irreducible matrix
	representations (irreps) after mapping with a vector.
          
algebraic loop. (Abbreviated to loop in these notebooks.) An m×m quasigroup Cayley index-table with a 1. 
           This is a multiplication table for m unsigned elements and for m-element vecs. It converts to an 
           r-signed algebra with m/r  r-signed elements if it can be folded over a central cyclic subgroup Cr.
           
annihilator.	A non-trivial vector with all sizes zero. E.g.{1,1,1,1,-2,-2} in S3. See also selector.

anti-commutative The property (possessed by Clifford & geometric algebra elements), that ab=-ba.

arcTanh[a,o] A generalization of Mathematica's ArcTanh[a/o] that gives a complex hyperangle depending
	          on the a,o (adjacent,opposite) octant. Many mathematicians use (opposite, adjacent).
	          
binegation.  Ordinary (unary) negation -A, with -(-A)=A, resulting from negation by C2 equivalencing, and
	         arising if the hidden axiom “minus one exists” is valid. Often incorrectly conflated with
	         subtraction (the r=2 case of subtractive elimination) which is a binary operation B-A.
	         
bivector       (Subscript[e, i])⋀(Subscript[e, j]), the wedge product of two distinct monovectors, an element of a Geometric Algebra.
                    It represents a directed area, but carries no shape information. Rotations occur in such areas.
                    
C2, C3 etc., The cyclic groups (associative loops) of 2,3, etc. primal elements. Do not confuse with Rings (Subscript[ℤ, 2])
  	       etc, for which negation is defined. Negation is often wrongly assumed for loops and groups.
  	       
C8xC2 etc.   The direct composition of C8 with C2 etc. Indirect composition is shown as e.g. C8iC4, C8jC4.
	       In GAP, indirect composition is (ambiguously) shown as C8:C2.
	       
C3	       Ternal, Terplex; the algebra of three elements, equivalence relationships on C3xC2, C3xC4 etc.

Cartesian- A vec or vector as an indexed set of coefficients with related directions or dimensions. Abelian
Form.         algebras have dual Polar Forms (q.v.) with offsets, radii, and angles. See toPolar, toVector.

Clifford        The anti-commutative sub-set of hoops, CLpq, created by cLoop[p,q], with 2^(p+q) elements, 
(geometric)  with p anti-commuting generators squaring to +1 and q squaring to -1. Other orderings are
algebras.      isomorphic to one of these. The elements partition into a scalar (first), pseudoscalar (last), p+q
	          relators (monovectors), and bi-, tri, etc k-vectors. A k-blade is a a set of elements of one 
	          dimensionality or grade. A multivector is a set (often sparse) of coefficients. The
	          non-Abelian geometric product, ab=a.b+a⋀b, extends to multiple dimensions, unlike the
	          related dot and cross products.  ab=a.b+a∧b is sometimes used to define geometric algebra in
	          terms of the dot and wedge products, but corresponds to the Jordan & Lie products for any
	          non-commutative hoop (not just for Cliffords). 
	          
CLpqr      My extended Clifford-like algebras. CLpqr have 3-fold symmetry if r=3, and are based on non-
	      associative octonions or split-octonions if r=1 or 2.    
	       
cTimes[A,B,o]  The hoopTimes multiplication procedure, modified to give AB, A.B, and A⋀B etc. for
                     Cliffords. The o parameter is used to give different products (geometric, Lie, Jordan, etc.).
                     
collapse.      Used in earlier versions, now replaced by fold, replacement of r-fold symmetry by r-signs.

conservative.The property (discovered by Frobenius for group inverse tables) of some multiplication tables
	         to conserve the determinant on vector multiplication. The defining property of hoop
	         vector-division algebras - many signed tables, all groups, and some non-associative
	         octonionic Moufang loops define conservative algebras. Generalised-conjugate determinant
	         factors are conserved. Gerrard (2010) showed that these were “irreps”.
	         
conformal      Spaces with signature p,q are described as null vectors in cl[p+1,q+1]. Points, lines,
  geometry.    circles and spheres are geometrical primitives in conformal geometry.
  
det.	Generalised Determinant. (Handles rectangular and degenerate tables.)  The primal parent of
            Determinant, Permanent, and Moore-Penrose inverse. det[a_?MatrixQ, r_:2, t_:Times] calculates 
            a generalized determinant, as a list containing m components, for a (not necessarily square)
            matrix. Putting ‘r’ ={1,2,3,4,5} gives the {permanent, ditermanent, tritermanent, teterminant,
            penterminant}; the traditional Determinant is the signed sum of the two ditermanent terms. 
            In effect, each element of ‘a’ is multiplied by a power of the sign ‘𝕤’, with 𝕤^r=1; the final
            result comprises the signed sums of all the determinant-like terms with a given power of ‘𝕤’.
            The calculation is recursive via a stripped down version, ‘det1’, a signed row or column
            matrix being the basis. Short matrices are transposed to become ‘long’, i.e. with more columns
            than rows. ‘det’ sums all the square det’s, as in the Moore-Penrose inverse.
            Replacing ‘t_:Times’ by another binary operation creates new functions.
            
(Subscript[d, i])          A basis direction for a loop. (Subscript[x, i]) is a basis dimension, folded from 2 or more directions.
	 Directions can only be associated with operators or primal coefficients.
	        
dozal, doz.  The algebra of 12-element directors and vecs, and the unital algebra derived from it.

dual.	    Polar Duals exist in Abelian hoops as a set of offsets, squared radii or ulnae, and angles.
	     Clifford Duals result from exchanging k grades with p+q-k grades, and are not polar.
	     
D6,D8, etc. Dihedral groups, symmetries of the triangle, square, etc. Written S3, D8 when used as hoops.
 	        WikiMaths says that algebraists prefer D2n (as here & in GAP) but geometers prefer Dn.
 	        
element. (Subscript[a, 1])(Subscript[d, 1]), (Subscript[b, 2])(Subscript[x, 2]), c etc. A constituent part of a vec or vector, consisting of a coefficient ((Subscript[a, 1]), (Subscript[b, 2]))
	     and a direction or dimension ((Subscript[d, 1]), (Subscript[x, 2])) (often implied by the index or the position in the set).
	     
equivalencing.  The creation of a signed hoop table or vector by folding, an equivalence relationship
	     between sets (pairs, triples, quads etc) of primal elements (Subscript[r, i])~((Subscript[a, i])-(Subscript[a, i+m/r]))(Subscript[x, i]). Impossible for
  	     some loops, such as C3 & S3, which can only be folded after first composing with C2, etc.
	     Complex hoops and vectors are folded from quads of primal elements, i.e. r=4. Generalized or 
	     extended signs are distinct powers of roots of unity, created by folding. (See fold.)
	     
extended signs. Generalizations of +,-,+i,-i; e.g. (𝕤^𝕣), (𝕟^9), (𝕠^8), (𝕧^7), (𝕙^6), (𝕡^5), (𝕏^4). (𝕐^3), (𝕁^3), (𝕜^2), (𝕞^2) all = +1

(Subscript[f, j])	  The j'th distinct factor of the determinant of G. It is conserved, if G is a hoop, on multiplication;
	   if it is linear (L1) it is also conserved on addition. Distinct factors are sizes for the hoop algebra.
	   The list of sizes is the shape of a vector. If GAPdb is available, it can be found from smat.
	   
faithful representation. Sets of matrices that act as elements of a specified group. Created by ma as 2x2,
               3x3, or 4x4 unitary monomial matrices from basic sets chosen from matrix lists u2, u3, or u4.
               
flat vector. The usual representation of a vector as a flat list. cf sparse, graded.

fold.    See Equivalencing, signed table, multi-fold. The conversion of a loop with m unsigned elements to
           a hoop with m/r signed elements, generating a signed multiplication table. All groups can operate a
           algebras over fields because any group X can be considered to be folded from XF, where F is a field
           group such as C2 (real) or C4 (complex). Vecs (unsigned sets) fold to generalized signed vectors.
           If GAPdb is available, fromdb[m,n,o] creates the hoop folded over o-1 central subgroups.
            
fragment. A linear term in a compact size expression e.g. (a-b) in ((((a-b))^2)+(((b-c))^2)+(((c-a))^2))/2; also the
	     quadratic terms in compact expressions for quartic sizes. As they always occur as squares, they
	     are "symmetric differences" that do not involve subtraction (despite the minus sign!). The entries
	     in irreducible matrix representations are fragments.
	     
Frobenius conservation. The property that Det[A] Det[B]=±Det[AB] when vectors are multiplied using
	   some Cayley Tables. Det is the conjugate of that of the Cayley table mapped with the vector
	   elements. Frobenius proved that all associative tables (groups) are conservative. I found that
	   Octonions and a few other non-associative Moufang loops are conservative, and the property
	   survives on folding to a signed table. As with fragments, the ± sign indicates a "symmetric
	   difference". For real tables, conjugation has no effect, so the sizes are factors of the determinant.
	   
GAP      The immense Groups, Algorithms, Programmes package, which does not handle real or complex
	   numbers, but uses roots of unity extensively. The source of most of the ALGHC and GAPdb data.
	   
Gaussian Pulse.  Multi-dimensional Exp[go + i ω T+ (∑_(x,y,..))[i kx X - jx²(X+xi-vx T)²].

G:(Subscript[d, i])×(Subscript[d, j])→ (Subscript[d, ij]) A finite loop, group, hoop or groupoid giving the direction or basis dimension (Subscript[d, ij]) resulting
	   from the multiplication of (Subscript[d, i])&(Subscript[d, j]). Det[G] is that of the symbolic Cayley Table.
	   
generalized conjugate. An involute set that may give the conserved determinant factors. Real sets are their
	own conjugate. The terplex conjugate of {a,b,c} is {a,c,b}. The complex conjugate of {r,i} is {r,-i}.
	
generalized negation. Equivalence relations on sets of r primals (Subscript[n, i])~{(Subscript[a, i]), (Subscript[a, i+m/r]),...(Subscript[a, i+m(r-1)/r])}.
	    Subtraction is 2-fold additive elimination.
	    
generalized signs. Generalizations of +, -, +i, -i; with (𝕤^𝕣)=+1. See "Extended Signs" for a proposed standard.

gi, gp       Lists created by sd, used by hoopInverse, cInverse & toPolar. 

gmmnn   A mnemonic for groups given by GAP>SmallGroup(mm,nn), or loops in the extended GAP Atlas.

grade. In Clifford (geometric) algebras, the number of monovectors in the dimension of an element. Scalars
	are grade 0; bivector elements are grade 2 and may represent an oriented area; trivector elements
	are grade 3 and may represent an oriented volume. tocGr[A] converts a flat vector into a list of
	elements grouped by grade; tocGr[A,k] implements <A>k, selecting every element of grade k.
	
graded vector. A vector stored as a list of lists of increasing grade. cf flat, grade, sparse.

hoopInverse[A,hoop] The hoop left-multiplicative inverse Ai, permitting division in the specified hoop;
           AB/A= Ai·AB=B. Calculated as sums of derivatives of shapes wrt "specific variables" that are fixed
           by the location of the 1's in the index table. In signed tables, each term has to be multiplied by the sign
           of each 1. The location and multiplier are supplied as gi in the shape of the hoop. This is a Cramer's
           rule calculation, so the inverse has the determinant as a divisor. When this factorises, the inverse
           splits into partial-fractions with each size becoming a denominator. Division is left-multiplication by
           an inverse. Multiplication and division both eliminate infinite partial-fractions (created by zero sizes)
           by ejecting remainders, giving a renormalised result matching a pseudo-inverse. The result is
	projected onto a constrained sub-algebra. cInverse is the same, but A can be sparse or graded.
	gapInverse uses smat and avoids the inversion of the full Cayley table.
	
hoopPower[A,power]  A procedure to convert a vector to a power or root. It raises offsets and radii to the
	 required power, and multiplies angles by the power. The target hoop polar form must be known.
	 If angles exceed 2π they "wrap round", and then ((((A^p)))^(1/p)) will be a rotated version of A. Not
	 defined for non-abelian hoops, which may introduce uncertainty.
	 
hoopTimes[A,B,hoop]   The hoop multiplication rule, multiplying vectors A & B according to the specified
	   hoop. If the multiplicands have disparate zeroes, sizes that become zero in the product are
	   ejected into remainders to maintain size conservation. The product is then projected into a
	   sub-algebra. cTimes, gapTimes are multiplications that ignore remainders.
	   
hoop.   An algebra (H,+,·,\,/,1) with addition +, Moufang ·, left \ & right / division, 1, generalized signs,
	 and the Frobenius determinant conservation property. All groups, a few non-associative
	 Moufang Loops, and signed tables folded from them, are Hoops. They are in the database as
	 Protoloop preferred Cayley table isomorphs. Loops with names ending in n are not hoops.
	 
hoop algebra. Multiplication and division of vectors using a hoop multiplication table, with addition and
	    generalized subtraction. Powers & roots exist in Abelian hoop algebras.
	    
helix identity.  (∑_(n=1)^(n=m))((Sin[t+2n π/m])^(2p))=m Binomial[2p,p]/(4^p) (p<m/2 if m is even, else p<m)
	   The generalisation of  (Sin^2)[x]+(Cos^2)[x]=1. Derived by J.J.Thwaites (personal communication).
	   
hexal, hex. The algebra of 6-element directors and vectors, and the unitary algebra derived from it.

Hub. 	(provisional term). non-Abelian (non-commutative) hoop vectors have inverses that consist of a
	hub or central vector, and a set of null-vector spokes. These introduce uncertainty into division.
	
hyperbolic,  The analog of polar, where a size includes a difference of 2 squared fragments. The Klein
  Hypolar.     group is the archetype, with squared "ulnae" (a±c)^2-(b±d)^2 and angles arcTanh[a±c,b±d].
  
𝕙,𝕁,𝕜,𝕞 etc.  Generalized signs, different roots of unity. Proposed usage in Extended Signs. 'h etc. in ASCII.

i	        ((+1)^(1/4)), usually written ((-1)^(1/2)), a complex unit element. Ambiguous in Dav, CLpq & C4C4. etc.

incantation. My name for the expressions used to create a Cayley table using any of the procedures ca, cd,
	         cl, co, fromdb, ge, ma, md, ms, pe,sgroup, or ts.
	         
irreducible matrix representations	 (irreps).
        
isomorphs, isotopes I define isomorphs as loops with renamed (shuffled) elements whilst isotopes are
          signed loops with some changed signs that conserve the structure, but not the details, of the shape.
          
involutions. See Plex-conjugates. Vectors with changed signs of some elements. [Lounesto p29]

indexed set. A set in which an index identifies each element, so that permuted sets are set isomorphs.

Jordan product. The Abelian component of a non-commutative product, a.b in  ab=a.b+a⋀b, the inner
                or dot product. Non-associative. See Clifford Algebras.
                
K	   The Klein four-group, C2xC2, and the corresponding algebra, which has three square roots of
	   +1,  and equivalences to a "double algebra" with two. See Hyperbolic, Study numbers.
                
KdVB	     Korteweg-deVries broadened equation. [ n(1+n r)(A^(1+2/n))(Subscript[A, x])+(r-1)((Subscript[A, x])^2)/A(Subscript[ +(Subscript[A, x x])], x]) +
	     (Subscript[[(A^(1+m))(n^2)(r+m)],  t]) =0. Some multi-dimensional cases have travelling pulse solutions.
	     
large number  The observation that several ratios in physics are (approximately) low powers of (10^40); 
  hypothesis.     expressed by Dirac as (e^2)≃(10^40)(Subscript[G, m])t. It may be related to multi-radius polar-orbits. 
  
L1, L2, etc. Linear (L1), quadratic (L2) etc. sizes, conserved in an algebra, and given by shape.  L26ζλϕ is
	       the quadratic (L2) size which splits into six squared fragments and contains ζ, λ, & ϕ in its polar
	       form. L1 sizes are conserved on addition, all sizes are conserved on multiplication for hoops.
	       Originally misnamed "lengths"; the n'th root of Ln may be a length, and may provide a metric.
	       
length.      A constituent of a Polar, corresponding to either a scalar or, if the n'th root of Ln, to a radius or
	      ulna in polar or hypolar coordinates. (The other constituents are angles and offsets.)
	      
Lie product. The non-Abelian component of a non-commutative product, a⋀b in  ab=a.b+a⋀b, the outer
                 or dot product. Non-conservative. Possesses the Jacobi property, so it is a Lie algebra.
                 
limiting   The non-linear partial differential equation that has any fixed-shape fixed speed wave, travelling
 velocity  in any direction in d spatial and 1 time direction, as a solution. Massive and light-like particles arise
 eq'n.       if pairs or directions combine to give dimensions, or if some directions are "rolled-up" (Kaluza
	     -Klein). Gtt-∑[Gxx]=0 (with Gxx=(d^2)G/d(X^2) etc. and summation is over x1...xd).
	     
loop.	   An algebraic loop. An m×m quasigroup Cayley table with a 1, for a set with m unsigned elements.
                See Moufang Loop. Protoloops include signed tables as well as strict (unsigned) loops.
                
mm	   The number of primal elements in a loop, group or director. Hoops have mm/r signed elements.

monovector. My name for the generators (aka relators, gammas) of groups, Cayley-Dickson & Clifford
	   algebras. The unit is scalar, products are bivectors, trivectors, etc. Commonly and confusingly
	   called vectors  in the Geometric Algebra literature.
               
Moufang Loop. An m×m multiplication table for a set with m elements with L & R inverses (Subscript[L, i])(Subscript[R, i])=(Subscript[R, i])(Subscript[L, i]) and
	      the Moufang properties, including zx.yz=(z.xy)z, relaxed to "up to a sign" for signed tables. All
	      groups are Moufang Loops. A few non-associative Moufang Loops (e.g. octonions) are hoops. 

Multifolding. A loop with one or more central cyclic subgroups can be folded on each. Abelian hoops can
		be folded all the way down to C1 (in a single step for prime groups). Folding different
		isomorphs may produce an unsigned algebra, but will usually require the intoduction of
		powers of the r’th root(s) of  unity as generalised signs.
		
multi-phase   (∑_(n=1)^m)c Cos[ωt+2n k π/m+ϕ](Subscript[ζ, n])+f[t], a solution to (Subscript[y, n])" = (a^2)((Subscript[y, n-1])-2(Subscript[y, n])+(Subscript[y, n+1]))+f"[t] or 
   sinusoid.	(Subscript[y, n])' = a((Subscript[y, n-1])-(Subscript[y, n+1]))+f'[t].  ((Subscript[e, r])^(𝕤 x))generalizes (e^(i x))to define them.
   
multivector.   The Clifford (geometric) algebra name for a general indexed set of elements or blades.

non-commutative. The non-Abelian property that ab≠ba for at least one pair of elements.
	     Non-commutative  hoops have repeated determinant factors and so cannot revert polar
	     duals to the Cartesian form.
	     
n-potent.  A vector which is its own n'th power in a particular algebra, (A^n)=A. An Idempotent if n=2.

OC311.    A typical continuous polar-orbit, being the C3 orbit with both sizes equal to 1.

octr, octi. The non-associative 8-element Octonion & split-Octonion algebras. Created by cd[-1,-1,±1]
                  &, cLoop[p,q,1 or 2]. 2-folded from Oct and Q8M.
                  
offset.   The L1 sizes after conversion to polar form. See Radius, Polar Form. Symbols α, β, γ, δ.

partition. One of the monosized vectors (Subscript[B, n]) that sum to a vector B. Created by multiplication by a
	  projector. The inverse is obtained by summing 1/(Subscript[B, n]).
	  
Pauli-σ algebra. The algebra defined by the Pauli-σ 2×2 matrix multiplication tables.
	Conserves (t^2)-(x^2)-(y^2)-(z^2).

phase.	 One or more cyclically varying element in an orbit. A phase has a phase angle and a radius or ulna.

plex-conjugate. The generalization of complex-conjugate to some hoops, a vector with some elements
	  negated or (in ternary algebras) with 𝕁↔(𝕁^2) exchanges. Elements of the product of the vector and the
	  conjugates (which are involutions) are related to the sizes of the vec, often as signed sums of squared
	  elements. The generalised complex-conjugate plex is supplied (if known) in sd for a hoop.
	  
polar form.  A Dual representation of a vector as scalars (displacements), radii and angles, generalised
	polar coordinates.

polar angle. An angle parameter in a polar form or orbit. Symbols σ, τ, ϕ, φ, χ, ψ.

polar-orbit. A unital parametric representation of a director or vector, with multi-phase sinusoidal elements
	  depending on at least one free angle parameter {σ,τ..}. Greek letters α…κ are used as the
	  size parameters. An example is given under Size. Putting τ = -((Subscript[E, p]) t-P.x)/ℏ  gives the orbit the
	  properties of a multi-phase deBroglie wave. (Related to the “discrete orbits” in a group.)	  

polyhelix identity.  (∑_(i=0)^(i=m-1))(((Sin[t+2iπ/m]))^(2p))= (4^(-p))m (({{2p}, {p}})), {p<m if m is even, else p<2m}. See Helix.
powerSinusoid.   v=((Sin[ωt+ϕ])^p), which is a solution to (Overscript[v, ¨]) +(1/p-1) (((OverDot[v]) )^2)/v+p (ω^2)v=0. The travelling
	        wave V=(a Sin[(∑_(x,y,..))kx*(x+xi-jx*t)])^p is a solution to Vtt+(1/p-1) (Vt^2)/V-pV(∑_(x,y,..)) 
	        
primal,  𝔸.  Non-negative continuous, Absolute, half-line numbers. Real and Complex numbers are 
	   equivalence-relationships on pairs or quads of primal numbers. Other equivalence relations
	   on r primals give Terplex numbers (r=3) etc.
	   
protoloop. A prefered isomorph of a table, showing any structure; shapes are defined for the protoloop. The 
	  makeProtoloop (mp) procedure generates the Cayley table (with options to create some isomorphs).
	  The canonical form is the “smallest” isomorph when flattened to an ordered set.
	  
projector. A vector (Subscript[A, n]) that projects any vector onto a monosized partition (Subscript[B, n]): (Subscript[A, n])*B = (Subscript[B, n])+(Subscript[R, n]). ((Subscript[R, n]) is a
	remainder.) It is a 2-potent vector ((A^2)=A) that commutes with everything. Aka sub-space identity.
	
projective geometry. Spaces with signature p,q are described as null vectors in cl[p+1,q]. This has
	         points and line as geometrical primitives.
	         	
pseudo-root.    Non-abelian polar duals involve non-conserved size parameters and angle that do not add on
	   multiplication, but which have valid reversions to the cartesian form. When used to calculate roots
	   and powers, arbitrary angles and uncertainty are introduced, giving rotated pseudo-roots.
	   
pseudoscalar.  ℑ, the highest grade in a Clifford algebra, (Subscript[e, 1])⋀(Subscript[e, 1])⋀...(Subscript[e, n]), which replaces i in many
              algebras, but which anti-commutes with other elements if n is even.
              
radius.   The length associated with an angle in a polar representation. Symbols ϵ, ζ, η, κ, λ. See Ulna.

rational pulse. v=(((a/(1+(∑_(x,y,..))jx(xi+x-kx t)^2)))^n) 

relator.  The GAP term for a generator, gamma, or monovector, a primitive root of unity used to define
               a table in terms of powers and relationships.
               
remainder. Multiplication (or division) by vectors with zero-sizes creates left and right "remainders" Rr, Rl
	with the sizes that are present in the multiplicands but lost from the product, ensuring that hoops are
	conservative and "division by zero" is eliminated. A*B→AB+Rr+Rl, AB/B+Rr=A, AB/A+Rl=B.
	
𝕤         The generalised sign or 𝕣'th root of unity, (𝕤^𝕣)=1, often incorrectly treated as a point on the complex
	unit circle, but best considered as a direction. I write it as  's in printable ASCII and in irreps or smats.
	
selector.  A vector with one size=1and the rest=0 in a particular hoop, that commutes with all elements.
	Multiplying a vector by a full set of selectors creates a partition.
	
sh[mnemonic]  The symbolic determinant factors of the named loop, for use with genInverse, gi and gp
	 for use with hoopInverse, toPolar, toVector. It includes a plex for many small hoops.
	 
shape. The sizes of a director or orbit, found by shape[vec,mnemonic]; needed by genInverse, toPol etc.

shape[{ø_,a_,b_...}]  Factor (size) data for the target hoop, supplying the shape of {ø,a,b,..} as a list of individual
	   sizes {(Subscript[f, 1]),…(Subscript[f, k])} & setting gi, gp for genInverse, toPolar. It also sets plex for many small hoops.
	   
signed table. A multiplication table for a set which forms a group or loop on appending its signed elements.
	The multiplication table is the upper left quarter (ninth, sixteenth, etc.) of the full Cayley table, with
	 (possibly signed) elements folded, by equivalence relationships, from the full table elements.
	 
size.	The value of a single conserved property of a director or orbit, for a given hoop algebra. Thus the
            polar-orbit OC3αγ={α+2γ Cos[τ],α+2γ Cos[τ+2π/3 ],α+2γ Cos[τ+4 π/3]}/3 has individual sizes
            L1α=α, L23ζ2=(ζ^2), and so has a shape {α,ζ}, in the algebra C3; (hyper)complex-conjugate
            Eigenvector pairs are sizes factorized into complex pairs (etc) but are not always conserved. Sizes
            are always symmetrical ideals, and may fragment to compact expressions. Irreducible matrix
            representations become sizes when mapped with vectors
	
spin. 	A property of an orbit, which is reversed by reversing the sequence of the phases. Clockwise spin 
	 (Cos, -Sin, -Cos, Sin) is taken to be positive or Up.
	 
splitting.  The primal vector equivalent of subtraction, Split[A,B]=C if B+C=A. See SymDiff.

Spoke.  (provisional term)  See Hub.

Study numbers.  a+𝕜 b, (𝕜^2)=1, folded C2kC2, the algebra of the hyperbolic plane. Louenesto 2001 p24.

symDiff.  Unsigned difference between numbers s=|a,b| : s+Min[a,b]=Max[a,b]. Primal "additive inverse".

tocGr.     Converts a flat vector to a graded vector. tocGr[A,k] selects the k'th grade of A, i.e. <A>k

toPolar[v,h].    The procedures to convert cartesian representations v into polar forms p using hoop h.

toVector[p,h].	The procedures to convert polar representations p into cartesian forms v using hoop h.

ternegation.	-(-(-A))=A in systems with ternary symmetry, instead of -(-A)=A in simple (bi)negation.

terplex.  The algebra of the C3 table over the real or complex field.

tetral.    The algebra of the C4 and K tables over the real or complex field.

triality.  (Louenesto p309) 'sends a simple rotation to a positive isoclinic rotation and a positive isoclinic
             rotation to a negative isoclinic rotation'. The Moufang identity results in a special case of Cartan's
             triality. See 'Loops' in the GAP package.
             
trivector   (Subscript[e, i])⋀(Subscript[e, j])⋀(Subscript[e, k]), the wedge product of three distinct monovectors, an element of a Geometric
                  Algebra. It represents a directed volume, but carries no shape information.     
                        
ulna.	The hyperbolic analogue of radius. (u^2)=(x^2)-(y^2).

unity.   The object {1,0,…,0}  in unitary algebras, which leaves any object unchanged after multiplication.
	 Not to be confused with the identity matrix. The product of a vector and its inverse is unity.
	 
unit.     An object (e.g. {2/3,-1/3,-1/3} in C3) having all sizes 0 or 1 in a particular algebra.

unital.	A vec or vector with sizes that multiply to 1 in a specific algebra, ignoring zeroes, is unital. Orbits
	 are unital. (Clashes with "unital algebras", which are monoids and include all hoops.)
	 
vec.      An ordered list of m primal elements. Folds to an m/2 real, m/3 ternary or m/4 complex vector,
	 in algebras with 2, 3, or 4-fold symmetry.
	 
vector-division algebra.	A vector space with an invertible bilinear vector-multiplication operation
	and an	addition operation with additive elimination. The multiplication table will have the
	Moufang division property, so every vector has a multiplicative inverse; if it is folded over a
	central C2 or C4 subgroup the addition will have subtraction as its inverse.
	 
α,β,…λ  Size parameters in an orbit, also the corresponding length parameters in a Polar Dual.

(Subscript[x, i])	  The i'th basis dimension. Two or more directions may fold to a real dimension (Subscript[x, i])~{(Subscript[d, i]),(Subscript[d, i+m/2])}. 

σ,τ,…,ψ  A frequency parameter in an orbit. Additive in abelian multiplication.Also an angle parameter
                in a Polar.