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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.



Boolean algebra.
Axioms

# ref: Stephen Wolfram, A New Kind of Science

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

function NAND(2)

variable x,y,z

axiom ∀x ∀y ∀z NAND NAND NAND xy z NAND x NAND NAND xz x =z # (((xy)z)(x((xz)x)))=z

Models
model 1_1
 NAND
   0
model 2_1
 NAND
   1 0
   0 0
model 2_2
 NAND
   1 1
   1 0
model 4_1
 NAND
   3 2 1 0
   2 2 0 0
   1 0 1 0
   0 0 0 0
model 4_2
 NAND
   3 2 3 2
   2 2 2 2
   3 2 1 0
   2 2 0 0
model 4_3
 NAND
   3 3 1 1
   3 2 1 0
   1 1 1 1
   1 0 1 0
model 4_4
 NAND
   3 3 3 3
   3 2 3 2
   3 3 1 1
   3 2 1 0
      
Axioms

# ref Huntington 1904, https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Axiomatics

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

function A(2,infix) #AND

function O(2,infix) #OR

function N(1) #NOT

constant 0,1

variable x,y,z

axiom ∀x xA0=x #identity O

axiom ∀x xO1=x #identity 1

axiom ∀x (xONx)=1 #complement

axiom ∀x (xANx)=0 #complement

axiom ∀x ∀y xOy=yOx #commutativity of O

axiom ∀x ∀y xAy=yAx #commutativity of A

axiom ∀x ∀y ∀z xO(yAz) = (xOy)A(xOz) #distributivity

axiom ∀x ∀y ∀z xA(yOz) = (xAy)O(xAz) #distributivity

Axioms

# ref Huntington 1933

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

function +(2,infix)

function n(1) #complement

variable x,y

axiom ∀x ∀y x+y=y+x #commutativity

axiom ∀x ∀y (x+y)=y+x #associativity

axiom ∀x ∀y n(n(x)+y)+n(n(x)+n(y))=x #huntington eqn

Axioms

# ref uses Huntington 1933 with huntington's eqn replaced by its dual

# Robbin's conjecture proved by McCune, see https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Axiomatics

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

function +(2,infix)

function n(1) #complement

variable x,y,z

axiom ∀x∀y x+y=y+x #commutativity

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

axiom ∀x∀y n ((n(x+y)) + (n(x+ny)))=x #Robbins eqn

Comments

Certain models of size 2 are used to build computers.




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