Definitions and Propositions pdf
Part A: Order Theory 2
Definition A.8 - Bounds, Infimum, Supremum
A = (A, ≤) is a poset, B ⊆ A is a subset of A. An element a ∈ A is called upper bound of B, if b ≤ a for all b ∈ B. An upper bound a of B is called least upper bound of B or supremum of B (sup B), if a ≤ a' for all upper bounds a' of B.
An element a ∈ A is called lower bound of B, if b ≥ a for all b ∈ B. A lower bound a of B is called greatest lower bound of B or infimum of B (inf B), if a ≥ a' for all lower bounds a' of B.
The notation x ∨ y ("x join y") for sup{x,y}, and x ∧ y ("x meet y") for inf{x,y} is often used.
In the AWB an order is called algebraizable if and only if at least one of the operations meet or join is totally defined.
To illustrate Definition A.6 - Chain we have here a few remarks regarding the poset A = (A, ≤) which is visualised in figure A below:
{a, b, c} ⊆ A forms a chain. Elements b and d are incomparable, while elements a and f, for example, are comparable. The subset {a, b, c, d, e} contains no maximum, but instead there are two maximal elements: c and e. The elements c, e, and f are upper bounds of this set. Because there is no least upper bound the set {a, b, c, d, e} contains no supremum. (It does have an infimum: a). In A the element e possesses exactly one upper cover: element f.
Next to posets, lattices are another important construct; especially, but not only, in order theory.
Figure A: Two Hasse diagrams of the same posets
Definition A.10 - Lattice (cf. also Definition B.2 - Lattice)
A lattice is a poset A = (A, ≤) for which the following holds: For all x, y ∈ L there exists the supremum sup{x,y} and the infimum inf{x,y}.
Definition A.11 - Complete Lattice
A lattice A = (A, ≤) is called complete, if and only if any arbitrary subset X ⊆ A contains a supremum and an infimum.
According to Definition A.11 - Complete Lattice every finite lattice is complete.