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A note on starrish posets

by Victor Porton

78640, Shay Agnon 32-29, Ashkelon, Israel

Email: porton@narod.ru

Web: http://www.mathematics21.org

Abstract

In this note some theorems from my previous article are strengthened. Starrish posets, a

generalization of distributive lattices, are considered.

Keywords: distributive lattice

A.M.S. subject classiﬁcation: 06A06, 06D99

In this short note I strengthen some results about distributive lattices in [1] as distributive lattices

are a special case of starrish posets introduced in this note.

Deﬁnition 1. I will call a poset starrish when the full star ⋆a is a free st ar for every element a

of this poset.

Proposition 2. Every distributive lattice is starrish.

Proof. Let A is a distributive lattice, a ∈ A. Obviously 0

⋆a; obviously ⋆a is an upper set. If

x ∪ y ∈ ⋆a, then (x ∪ y) ∩ a is non-least that is (x ∩ a) ∪ (y ∩ a) is non-least what is equivalent to

x ∩ a or y ∩ a being non-least that is x ∈ ⋆a ∨ y ∈ ⋆a.

A generalization of theorem 1 in [1]:

Theorem 3. If A is a starrish join-semilattice then

atoms(a ∪ b) = atoms a ∪ atoms b

Proof. For every atom c we have: c ∈ atoms(a ∪ b) ⇔ c

a ∪ b ⇔ a ∪ b ∈ ⋆c ⇔ a ∈ ⋆c ∨ b ∈ ⋆c ⇔ c

a ∨

c

b ⇔ c ∈ atoms a ∨ c ∈ atoms b.

A generalization of proposition 30 in [1]:

Proposition 4. Let (A; Z) be a down-aligned ﬁltrator with ﬁnitely join-closed core, where A is a

starrish join-semilattice and Z is a join-semilattice. Then atomic elements of this ﬁltrator are prime.

Proof. Let a be an atom of the lattice A. We have fo r every X , Y ∈ Z

X ∪

Z

Y ∈ up a ⇔

X ∪

A

Y ∈ up a ⇔

X ∪

A

Y ⊇ a ⇔

X ∪

A

Y

A

a ⇔

X

A

a ∨ Y

A

a ⇔

X ⊇ a ∨ Y ⊇ a ⇔

X ∈ up a ∨ Y ∈ up a.

1

A generalization of theorem 43 in [1]:

Theorem 5. Let (A; Z) be a star rish join-se milattice ﬁltrator with ﬁnitely join-closed core which

is a join-semilattice. Then ∂a is a free s tar for each a ∈ A.

Proof. For every A, B ∈ Z

X ∪

Z

Y ∈ ∂a ⇔

X ∪

A

Y ∈ ∂a ⇔

X ∪

A

Y

A

a ⇔

X

A

a ∨ Y

A

a ⇔

X ∈ ∂a ∨ Y ∈ ∂a.

A generalization of theorem 65 in [1]:

Theorem 6. Let (A; Z) be a semiﬁltered down-aligned ﬁltrator with ﬁnitely meet-closed core Z

which is an atomistic lattice and A is a starrish join-semilattice, then Cor

′

(a ∪

A

b) = Cor

′

a ∪

Z

Cor

′

b

for every a, b ∈ A.

Proof. Cor

′

(a ∪

A

b) =

S

Z

{x | x is an atom of Z, x ⊆ a ∪

A

b} (use proposition 34 from [1]),

By the theorem 50 from [1] we have Cor

′

(a ∪

A

b) =

S

Z

(atoms

A

(a ∪

A

b) ∩ Z) =

S

Z

((atoms

A

a ∪

atoms b) ∩ Z) =

S

Z

((atoms

A

a ∩ Z) ∪ (atoms

A

b ∩ Z)) =

S

Z

(atoms

A

a ∩ Z) ∪

Z

S

Z

(atoms

A

b ∩ Z)

(used the theorem 3). Again using the theorem 50 from [1], we get

Cor

′

(a ∪

A

b) =

S

Z

{x | x is an a tom of Z, x ⊆ a} ∪

Z

S

Z

{x | x is an atom of Z, x ⊆ b} =

Cor

′

a ∪

Z

Cor

′

b (again used the proposition 34 from [1]).

Bib liography

[1] Victor Porton. Filters on posets and generalizations. International Journal of Pure and Applied Mathe-

matics, 74(1):55–119, 2012.

2