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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.



Lattices
Axioms

# axioms in terms of algebra

relation =(2,infix)       {a0==a1}

function M(2,infix)

function J(2,infix)

variable x,y,z

axiom       ∀x xMx = x                     #meet idempotent

axiom       ∀x∀y xMy = yMx                     #meet commutative

axiom       ∀x∀y∀z xM(yMz) = (xMy)Mz       #meet associative

axiom       ∀x xJx = x                     #join idempotent

axiom       ∀x∀y xJy = yJx                     #join commutative

axiom       ∀x∀y∀z xJ(yJz) = (xJy)Jz       #join associative

axiom       ∀x∀y xM(xJy) = x              #absorption

axiom       ∀x∀y xJ(xMy) = x              #absorption

Models
model 1_1
 M
  0
 J
  0
model 2_1
 M
  0 1
  1 1
 J
  0 0
  0 1
model 2_2
 M
  0 0
  0 1
 J
  0 1
  1 1
model 3_1
 M
  0 1 2
  1 1 2
  2 2 2
 J
  0 0 0
  0 1 1
  0 1 2
model 3_2
 M
  0 0 2
  0 1 2
  2 2 2
 J
  0 1 0
  1 1 1
  0 1 2
model 3_3
 M
  0 0 0
  0 1 2
  0 2 2
 J
model 3_2
 M
  0 0 2
  0 1 2
  2 2 2
 J
  0 1 0
  1 1 1
  0 1 2
model 3_3
 M
  0 0 0
  0 1 2
  0 2 2
 J
  0 1 2
  1 1 1
  2 1 2


      
Axioms

# axioms in terms of relations

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

relation ≤(2,infix)

variable x,y,z,join,meet

axiom ∀x x≤x #reflexive

axiom ∀x∀y x≤y ⋀ y≤x ⟶ x=y #antisymmetry

axiom ∀x∀y∀z x≤y ⋀ y≤z ⟶ x≤z #transitive

axiom ∀x∀y ∃meet meet≤x ⋀ meet≤y ⟶ ∀z z≤x⋀z≤y ⟶ z≤meet

axiom ∀x∀y ∃join x≤join ⋀ y≤join ⟶ ∀z x≤z⋀y≤z ⟶ join≤z

Models
model 1_1
 ≤
   1
model 2_1
 ≤
   1 0
   0 1
model 2_2
 ≤
   1 1
   0 1
model 2_3
 ≤
   1 0
   1 1
model 3_1
 ≤
   1 0 0
   0 1 0
   0 0 1
model 3_2
 ≤
   1 1 0
   0 1 0
   0 0 1
model 3_3
 ≤
   1 0 1
   0 1 0
   0 0 1
      



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