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17.8. INFINITE PRODUCT OF POSET ELEMENTS

247

Proof.

GR

Strd

Y

a

=

(

L

Q

A

up

L

Z

GR

Q

Strd

a

)

=

(

L

Q

A

up

L

GR

Q

Strd

a

)

=

(

L

Q

A

K

up

L

:

K

GR

Q

Strd

a

)

L

Q

A

K

up

L, i

n

:

K

i

6

a

i

L

Q

A

i

n, K

up

L

:

K

i

6

a

i

=

L

Q

A

i

n

:

L

i

6

a

i

=

GR

Strd

Y

a

(taken into account that

Q

A

is a separable poset).

Theorem

1297

.

Staroidal product is a completary staroid (if our posets are

starrish join-semilattices).

Proof.

We need to prove

i

dom

A

:

A

i

6

(

L

0

i

t

L

1

i

)

⇔ ∃

c

∈ {

0

,

1

}

n

i

dom

A

:

A

i

6

L

c

(

i

)

i.

Really,

i

dom

A

:

A

i

6

(

L

0

i

t

L

1

i

)

⇔ ∀

i

dom

A

: (

A

i

6

L

0

i

A

i

6

L

1

i

)

c

∈ {

0

,

1

}

dom

A

i

dom

A

:

A

i

6

L

c

(

i

)

i.

Definition

1298

.

Let (

A

i

;

Z

i

) be an indexed family of down-aligned filtrators.

Then for every

A

Q

A

funcoidal product

is multifuncoid

Q

FCD

(

A

)

A

defined

by the formula (for every

L

Q

Z

):

*

FCD

(

A

)

Y

A

+

k

L

=

A

k

if

i

(dom

A

)

\ {

k

}

:

A

i

6

L

i

0

otherwise

.

Proposition

1299

.

Q

Strd

(

A

)

A

=

h

Q

FCD

(

A

)

A

i

.