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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.



Groups.
Axioms

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

function ⚬(2,infix)

constant 0

variable x,y,z

axiom       ∀x∀y∀z (x⚬y)⚬z = x⚬(y⚬z)       #associative

axiom       ∀x x⚬0=x                     #identity

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

Models
model 1_1
 ⚬
  0
model 2_1
 ⚬
  0 1
  1 0
model 3_1
 ⚬
  0 1 2
  1 2 0
  2 0 1
model 4_1
 ⚬
  0 1 2 3
  1 0 3 2
  2 3 0 1
  3 2 1 0
model 4_2
 ⚬
  0 1 2 3
  1 3 0 2
  2 0 3 1
  3 2 1 0
model 4_3
 ⚬
  0 1 2 3
  1 0 3 2
  2 3 1 0
  3 2 0 1
model 4_4
 ⚬
  0 1 2 3
  1 2 3 0
  2 3 0 1
  3 0 1 2

      
Axioms

# these have no existential quantifier

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

function ⚬(2,infix)

function inv(1)

constant 0

variable x,y,z

axiom ∀x∀y∀z (x⚬y)⚬z = x⚬(y⚬z) #associative

axiom ∀x x ⚬ 0 = x #identity

axiom ∀x x ⚬ inv(x) = 0 #inverse

Models
model 1_1
 ⚬
   0
 inv
   0
model 2_1
 ⚬
   0 1
   1 0
 inv
   0 1
model 3_1
 ⚬
   0 1 2
   1 2 0
   2 0 1
 inv
   0 2 1
      



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