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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.



Strict Weak Orderings
Axioms

relation ≤(2,infix)

variable x,y,z

axiom       ∀x ¬ x≤x                     #irreflexive

axiom       ∀x∀y ((x≤y) ⟶ (¬ (y≤x)))       #asymmetric

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

axiom       ∀x∀y∀z ((¬x≤y) ⋀ (¬y≤x) ⋀ (¬y≤z) ⋀ (¬z≤y)) ⟶ (¬(x≤z) ⋀ (¬z≤x))       #transitivity of incomparability

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



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