Definitions and Propositions pdf
Part A: Order Theory 4
Before wrapping up the section on order theory and before moving on to definitions and propositions pertaining to universal algebras, we want to show you a nice application of critical and weakly critical pairs.
Definition A.18 - Critical Pair
P = (X, P) is a poset. (x,y) is a critical pair if the following conditions hold:
- x and y are incomparable
- a < x ⇒ a < y ∀a ∈ X
- b > y ⇒ b > x ∀b ∈ X
(x,y) critical pair ⇏ (y,x) critical pair.
Definition A.30 - Weakly Critical Pair
P = (X, P) is a poset. (x,y) is a weakly critical pair (also called subcritical pair) if the following conditions hold:
- x ≰ y
- a < x ⇒ a ≤ y ∀a ∈ X
- b > y ⇒ b ≥ x ∀b ∈ X
Every critical pair is also a weakly critical pair. In the following we call a weakly critical pair which is not critical proper weakly critical.
Definition A.32 - Maximal 0-1 Sublattice
L is a finite, distributive lattice (L ∈ DLfin) and M ⊆ L is a subset.
M is a 0-1 sublattice of L if M is a sublattice of L with 0L, 1L ∈ M.
M is a maximal 0-1 sublattice of L if M is a 0-1 sublattice and there is one 0-1 sublattice M' for which the following holds: M ⊆ M' ⊆ L ⇒ M'=M or M'=L.
Definition A.33 - ∨-, ∧-irreducible Element, J(L)
L is a finite distributive lattice. x ∈ L is ∨-irreducible (join-irreducible) if and only if x = y ∨ z ⇒ x = y or x = z (definition of ∧-irreducible (meet-irreducible) is analogous).
J(L) is the set of all ∨-irreducible elements of L without 0.
The dual set M(L) (Meet-irreducible Lattices) is defined as the set of all ∧-irreduzible elements of L without 0.
Due to the symmetric nature of lattices a definition of only J(L) is sufficient.
Proposition A.34 - Hashimoto Proposition
Maximal 0,1-sublattices of L correspond bijektively to the critical and weakly critical pairs in J(L).