Editing Laws of Form (section)

===Relation to magmas===
The ''primary algebra'' embodies a point noted by [[Edward Vermilye Huntington|Huntington]] in 1933: [[Boolean algebra (logic)|Boolean algebra]] requires, in addition to one [[unary operation]], one, and not two, [[binary operation]]s. Hence the seldom-noted fact that Boolean algebras are [[magma (algebra)|magmas]]. (Magmas were called [[groupoid]]s until the latter term was appropriated by [[category theory]].) To see this, note that the ''primary algebra'' is a [[commutative]]:
*[[Semigroup]] because ''primary algebra'' juxtaposition [[Commutative property|commute]]s and [[associative property|associates]];
*[[Monoid]] with [[identity element]] [[Image:Laws of Form - double cross.gif]], by virtue of '''J0'''.

[[group (mathematics)|Groups]] also require a [[unary operation]], called [[Group (mathematics)#Definition|inverse]], the group counterpart of [[Boolean algebra (logic)|Boolean complementation]]. Let [[Image:Laws of Form - (a).png|20px]] denote the inverse of ''a''. Let [[Image:Laws of Form - cross.gif]] denote the group [[identity element]]. Then groups and the ''primary algebra'' have the same [[signature (logic)|signatures]], namely they are both <math>\lang - \ -, \overline{- \ |}, \overline{\ \ |} \rang</math> algebras of type 〈2,1,0〉. Hence the ''primary algebra'' is a [[list of algebraic structures|boundary algebra]]. The axioms for an [[abelian group]], in boundary notation, are:
* '''G1'''. ''abc'' = ''acb'' (assuming association from the left);
* '''G2'''. [[Image:Laws of Form - ()a=a.png|80px]]
* '''G3'''. [[Image:Laws of Form - (a)a=().png|80px]].
From '''G1''' and '''G2''', the commutativity and associativity of concatenation may be derived, as above. Note that '''G3''' and '''J1a''' are identical. '''G2''' and '''J0''' would be identical if &nbsp;&nbsp;[[Image:Laws of Form - double cross.gif|25px]]&nbsp;=&nbsp;[[Image:Laws of Form - cross.gif|20px]]&nbsp;&nbsp; replaced '''A2'''. This is the defining arithmetical identity of group theory, in boundary notation.

The ''primary algebra'' differs from an [[abelian group]] in two ways:
*From '''A2''', it follows that [[Image:Laws of Form - double cross.gif]] ≠ [[Image:Laws of Form - cross.gif]]. If the ''primary algebra'' were a [[group (mathematics)|group]], [[Image:Laws of Form - double cross.gif]] = [[Image:Laws of Form - cross.gif]] would hold, and one of &nbsp;&nbsp;[[Image:Laws of Form - (a).png|20px]]&nbsp;''a''&nbsp;=&nbsp;[[Image:Laws of Form - double cross.gif|30px]]&nbsp;&nbsp; or &nbsp;&nbsp;''a''&nbsp;[[Image:Laws of Form - cross.gif|30px]]&nbsp;=&nbsp;''a''&nbsp;&nbsp; would have to be a ''primary algebra'' consequence. Note that [[Image:Laws of Form - cross.gif|20px]] and [[Image:Laws of Form - double cross.gif|25px]] are mutual ''primary algebra'' complements, as group theory requires, so that <math>\overline{\ \overline{\ \overline{\ \ |} \ \Big|} \ \Bigg|} = \overline{\ \ |}</math> is true of both group theory and the ''primary algebra'';
*'''C2''' most clearly demarcates the ''primary algebra'' from other magmas, because '''C2''' enables demonstrating the [[absorption law]] that defines [[lattice (order)|lattices]], and the [[distributive law]] central to [[Boolean algebra (structure)|Boolean algebra]].
Both '''A2''' and '''C2''' follow from ''B''{{'}}s being an [[ordered set]].