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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.



Rings.
Axioms

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

function       +(2,infix)

function       .(2,infix)

constant       0

variable       x,y,z

axiom       ∀x x + 0 = x                            #identity

axiom       ∀x∀y∀z (x + y) + z = x + (y + z)       #associativity

axiom       ∀x∀y∀z (x . y) . z = x . (y . z)       #associativity

axiom       ∀x∃y x + y = 0                            #inverse

axiom       ∀x∀y x+y = y+x                            #commutative

axiom       ∀x∀y∀z x.(y+z) = (x.y)+(x.z)              #distributive

axiom       ∀x∀y∀z (y+z).x = (y.x)+(z.y)              #distributive

Models
model 1_1
 +
  0
 .
  0
model 2_1
 +
  0 1
  1 0
 .
  0 0
  0 0
model 2_2
 +
  0 1
  1 0
 .
  0 0
  0 1
model 3_1
 +
  0 1 2
  1 2 0
  2 0 1
 .
  0 0 0
  0 0 0
  0 0 0
model 3_2
 +
  0 1 2
  1 2 0
  2 0 1
 .
  0 0 0
  0 1 2
  0 2 1
model 3_3
 +
  0 1 2
  1 2 0
  2 0 1
 .
  0 0 0
  0 2 1
  0 1 2

      



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