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Example 1.

There is a separable poset (that is a set with

?

being an injection) such that

?x

?y

does not imply

x

v

y

(where

v

denotes our order) for elements

x

,

y

of this poset.

Proof.

(with help of sci.math partakers) Consider

5

-elements poset dened by the following Hasse

diagram:

a

p

q

b

r

?p

=

f

p; a; b

g

,

?q

=

f

q; a; b

g

,

?r

=

f

r; b

g

,

?a

=

f

p; q; a; b

g

,

?b

=

f

p; q; a; b; r

g

.

Thus

?x

=

?y

)

x

=

y

for any

x

,

y

in our poset.

?a

?b

but not

a

v

b

.

1