A relational graph T is called weakly antisymmetric [;\Leftarrow:\Rightarrow;]
[;\bullet\quad\quad \forall_{\ x\ y}\ \left(\ \left(\left(x\ y\right) \in T\ \ \&\ \ \left(y\ x\right) \in T\right)\quad\Rightarrow\quad x=y\ \right);]
A relational graph [;T;] is called strongly antisymmetric [;\Leftarrow:\Rightarrow;]
[;\bullet\quad\quad \forall_{\ x}\ (x\ x) \notin T;]
The following two properties of a relation [;\mathbf{T} := (X\ X\ T);] are equivalent:
- [;\mathbf{T};] is an equivalence and it is weakly antisymmetric;
- [;T = \Delta_X;]
A partially ordered set, or a poset for short, is a preordered relation [;\mathbf{T} := (X\ X\ T);], which is (weakly) antisymmetric; thus, to write the same explicitly, a poset is characterized by the following three properties:
- transitivity;
- reflexivity
- weak antisymmetry
Posets admit a similar notion of sharp posets. First we define a sharp partial order relational graph [;S;] as transitive and strongly antisymmetric in the above sense:
[;\bullet\quad\quad\quad \forall_x (x\ x)\ \notin\ S;]
A relation [;(X\ X\ T);] is called a sharp poset [;\Leftarrow :\Rightarrow;] [;T;] is a sharp partial order relational graph.
Given relations [;\mathbf{S} := (X\ X\ S);] and [;\mathbf{T} := (X\ X\ T);] the following two equivalences hold:
- [;\mathbf{S};] is a sharp poset [;\Leftrightarrow;] [;\mathbf{T};] is a poset such that [;S = T \backslash \Delta_X;];
- [;\mathbf{T};] is a poset [;\Leftrightarrow;] [;\mathbf{S};] is a sharp poset such that [;T = S \cup \Delta_X;];
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