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Online Encyclopedia of
Mathematical Models.






Models:

boolean algebras, lattices, directed sets, equivalence relations,
graphs, directed graphs, bipartite graphs,
pre-orderings, strict partial orders, strict weak orderings, partial orderings, weak orderings, total orderings,
groups, rings, fields, racks, quandles, Tarski's HS Algebra,
more coming soon.



Racks.
Axioms

relation =(2,infix) {a0==a1}

function ⊲(2,infix)

variable x,y,z,w

axiom ∀x∀y∀z x⊲(y⊲z)=(x⊲y)⊲(x⊲z) #self-distributive law

axiom ∀x∀y∃z x⊲z=y ⋀ ∀w x⊲w=y ⟶ z=w

Models
model 1_1
 ⊲
   0
model 2_1
 ⊲
   1 0
   1 0
model 2_2
 ⊲
   0 1
   0 1
model 3_1
 ⊲
   2 1 0
   2 1 0
   2 1 0
model 3_2
 ⊲
   2 1 0
   0 1 2
   2 1 0


      
Axioms

relation =(2,infix) {a0==a1}

function ⊲(2,infix)

function ⊳(2,infix)

variable x,y,z

axiom ∀x∀y∀z x⊲(y⊲z)=(x⊲y)⊲(x⊲z) #left self-distributive law

axiom ∀x∀y∀z (z⊳y)⊳x=(z⊳x)⊳(y⊳x) #right self-distributive law

axiom ∀x∀y (x⊲y)⊳x=y

axiom ∀x∀y x⊲(y⊳x)=y

Models
model 1_1
 ⊲
   0
 ⊳
   0
model 2_1
 ⊲
   1 0
   1 0
 ⊳
   1 1
   0 0
model 2_2
 ⊲
   0 1
   0 1
 ⊳
   0 0
   1 1
model 3_1
 ⊲
   2 1 0
   2 1 0
   2 1 0
 ⊳
   2 2 2
   1 1 1
   0 0 0
model 3_2
 ⊲
   2 0 1
   2 0 1
   2 0 1
 ⊳
   1 1 1
   2 2 2
   0 0 0
      



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