
A u B 2 F , A 2 F _ B 2 F is equivalent to :(A u B 2 F ) , :(A 2 F ^ B 2 F ) is equivalent to
A u B 2 F , A 2 F ^ B 2 F .
We have the following commutative diagram in category Set, every arrow of this diagram is an
isomorphism, every cycle in this diagram is an identity:
hduali
ideals
:
hduali
free stars
mixers
:
filters
Figure 1. Diagram Υ
(where : denotes set-theoretic complement).
These isomorphisms are also order isomorphisms if we define order in the right way.
The above it is defined for lattices only. Generalizing this for arbitrary posets is straigthforward:
Definition 2. Let A be a poset.
• Filters are sets F without the greatest element of A with A; B 2F ,9Z 2 F :(Z v A ^ Z v B)
(for every A; B 2 Z).
• Ideals are sets F without the least element of A with A; B 2 F ,9Z 2 F : (Z w A ^ Z w B)
(for every A; B 2 Z).
• Free stars are sets F without the greatest element of A with
A; B 2 F , 9Z 2 F : (Z w A ^ Z w B)
A 2/ F ^ B 2/ F , 9Z 2 F : (Z w A ^ Z w B)
A 2 F _ B 2 F , :9Z 2 F : (Z w A ^ Z w B)
• Mixers are lower sets F without the least element of A with :9Z 2 F : (Z v A ^ Z v B) ,
A 2 F _ B 2 F or equivalently 9Z 2 F : (Z v A ^ Z v B) , A 2/ F ^ B 2/ F (for every A; B 2 Z).
Proposition 3. The following are equivelent: [TODO: With one side implications and requirement
to be upper/lower set.]
1. F is a free star.
2. 8Z 2 A: (Z w A ^ Z w B ) Z 2 F ) , A 2 F _ B 2 F for every A; B 2 A and F =/ P A.
Proof. The following is a chain of equivalencies:
9Z 2 F : (Z w A ^ Z w B) , A 2/ F ^ B 2/ F ;
8Z 2 F : :(Z w A ^ Z w B) , A 2 F _ B 2 F ;
8Z 2 A: (Z 2/ F ) :(Z w A ^ Z w B)) , A 2 F _ B 2 F ;
8Z 2 A: (Z w A ^ Z w B ) Z 2 F ) , A 2 F _ B 2 F :
2