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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.



Tarski's High-School Algebra.
Axioms

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

function +(2,infix)

function *(2,infix)

function ^(2,infix)

constant 1

variable x,y,z

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

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

axiom ∀x x * 1 = x                            #identity

axiom ∀x∀y x*y = y*x                            #commutative

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

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

axiom ∀x 1^x = 1                            #identity

axiom ∀x∀y∀z x^(y+z) = (x^y)*(x^z)              #distributive

axiom ∀x∀y∀z (x*y)^z = (x^z)*(y^z)              #distributive

axiom ∀x∀y∀z (x^y)^z = x^(y*z)              #composition of exponent

Models
model 1_1
 +
   1
 *
   1
 ^
   1
model 2_1
 +
   0 0
   0 0
 *
   0 0
   0 1
 ^
   0 0
   1 1
model 2_2
 +
   0 1
   1 0
 *
   0 0
   0 1
 ^
   0 0
   1 1
model 2_3
 +
   0 0
   0 1
 *
   0 0
   0 1
 ^
   0 0
   1 1
model 2_4
 +
   0 1
   1 1
 *
   0 0
   0 1
 ^
   0 0
   1 1
model 2_5
 +
   0 1
   1 1
 *
   0 0
   0 1
 ^
   1 0
   1 1
      



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