relation ≤(2,infix)

variable x,y,z

axiom ∀x∀y x≤y ⋁ y≤x               #total

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 .

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

relation ≤(2,infix)

variable x,y,z

axiom ∀x x≤x #reflexive

axiom ∀x∀y∀z x≤y ⋀ y≤z -> x≤z #transitive

axiom ∀x∀y x≤y ⋁ x=y ⋁ y≤x #trichotomy

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

oeis.org/A000670