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6. MATTERS RELATED TO TENSOR PRODUCT

26

The first easily follows from ˜

φ

i

=

φ

and the fact that ˜

φ

preserves binary joins.

The second easily follows from ˜

φ

i

=

φ

and that

φ

preserves

.

The third follows from the fact that ˜

φ

preserves joins.

Corollary

2046

.

The poset of prestaroids

preStrd

(

A

) is isomorphic to an ideal

(on a join-semilattice), provided that

A

is an indexed family of join-semilattices.

Proof.

preStrd

(

A

)

=

SepJoin

(

A

,

2)

=

F

(

Q

A

)

/

∼→

2

=

I

(

F

(

Q

A

)

/

).

FiXme

: Check below (especially posets vs dual posets) for errors.

Corollary

2047

.

preStrd

is a complete lattice.

Proof.

Corollary 515.

Corollary

2048

.

preStrd

is a filtered filtrator.

Proof.

Theorem 531.

FiXme

: Try to prove that

preStrd

is atomic and moreover atomistic (under

certain conditions). Other properties?