Corrections

_chapters/01_set.md

@@ -68,7 +68,7 @@ Of course if one is a valid answer, so can be zero. If we want a set of all *bla
 
 Note that a set is defined only by the items it contains, which means that there is no difference between the set that contains zero *balls* and the set that contains zero *numbers*, for instance. In other words, the empty set is *unique* set, which makes it a very special one. Formally, the empty set is marked with the symbol **∅** (so **B = W = ∅**).
 
-The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| A ⊆ ∅**)
+The empty set is a special one, for example, it is a subset of every other set (mathematically speaking, **∀ A \| ∅ ⊆ A**)
 
 We will encounter the empty set again.
 
@@ -286,7 +286,7 @@ The Power of Composition
 
 To understand how powerful composition is, consider the following: one set being connected to another means that each function from the second set can be transferred to a corresponding function from the first one.
 
-If we have a function **g: P → Y ** from set **P** to set **Y**, then for every function **f** from the set **Y** to any other set, there is a corresponding function **f ∘ g** from the set **P** to the same set. In other words, every time you define a new function from **Y** to some other set, you gain one function from **P** to that same set for free.
+If we have a function **g: P → Y** from set **P** to set **Y**, then for every function **f** from the set **Y** to any other set, there is a corresponding function **f ∘ g** from the set **P** to the same set. In other words, every time you define a new function from **Y** to some other set, you gain one function from **P** to that same set for free.
 
 ![Functional composition connect](morphism_general.svg)
 

_chapters/04_order.md

@@ -60,7 +60,7 @@ This is the law that to a large extend defines what an order is: if I am better
 Antisymmetry
 ---
 
-The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (**a ≤ b ⟺ b ≰ a**). 
+The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (or in other words you have **x ≤ y** and **y ≤ x** only if **x = y**). 
 
 ![antisymmetry](antisymmetry.svg)
 
@@ -71,7 +71,7 @@ Totality
 
 The last law is called *totality* (or *connexity*) and it mandates that all elements that belong to the order should be comparable - **a ≤ b or b ≤ a**. That is, for any two elements, one would always be "bigger" than the other. 
 
-By the way, this law makes the reflexivity law redundant, as it is just a special case of reflexivity when **a** and **b** are one and the same object, but I still want to present it for reasons that will become apparent soon.
+By the way, this law makes the reflexivity law redundant, as reflexivity is just a special case of totality when **a** and **b** are one and the same object, but I still want to present it for reasons that will become apparent soon.
 
 ![connexity](connexity.svg)
 
@@ -171,7 +171,7 @@ Like with the maximum element, if two elements have several upper bounds that ar
 
 ![A non-join diagram](non_join.svg)
 
-If, however, one of those elements is established as bigger than another, it immediately qualifies.
+If, however, one of those elements is established as smaller than the rest of them, it immediately qualifies.
 
 ![A join diagram](non_join_fix.svg)
 

dictionary.txt

@@ -0,0 +1,49 @@
+coproduct
+morphism
+coproducts
+morphisms
+preorders
+Preorders
+antisymmetry
+antisymmetric
+preorder
+Preorder
+semilattice
+Semilattice
+semilattices
+Semilattices
+meet-semilattice
+meet-semilattices
+poset
+posets
+Posets
+Birkhoff's
+superset
+monoid
+monoid-like
+monoids
+monoidal
+monoid's
+Monoid
+Monoids
+isomorphism
+isomorphisms
+Hasse
+Antisymmetry
+connexity
+linearly
+Heyting
+modus
+Modus
+ponens
+intuitionistic
+leit
+Z3
+Z2
+Z1
+abelian
+Abelian
+non-abelian
+Dih3
+composable
+forall