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2. DEFINITION

42

Corollary

2231

.

h⊥i

x

=

for Galois connections from a poset

A

with

greatest element to a poset

B

with least element.

FiXme

: Clarify.

Theorem

2232

.

If

A

and

B

are bounded posets, then

GAL

(

A

,

B

) is bounded.

Proof.

That

GAL

(

A

,

B

) has least element was proved above. I will demon-

strate that (

α, β

) is the greatest element of

pFCD

(

A

,

B

) for

αX

=

(

B

if

X

=

A

>

B

if

X

6

=

A

;

βY

=

(

>

A

if

Y

=

>

B

A

if

Y

6

=

>

B

.

First prove

Y

v

αX

X

v

βY

.

Really

αX

v

Y

X

=

A

Y

=

>

B

X

v

βY

.

That it is the greatest Galois connection between

A

and

B

easily follows from

proposition

2229

.

Theorem

2233

.

For every brouwerian lattice

x

7→

c

u

x

is a lower adjoint.

Proof.

By dual of theorem 154.

Exercise

2234

.

Describe the corresponding upper adjoint, especially for the

special case of boolean lattices.

2. Definition

Definition

2235

.

System of presides

is a functor Υ = (

f

7→ h

f

i

) from an

ordered category to the category of functions between (small) bounded lattices,
such that (for all relevant variables):

1

. Every Hom-set of Src Υ is a bounded join-semilattice.

2

.

h

a

i⊥

=

.

3

.

h

a

t

b

i

X

=

h

a

i

X

th

b

i

X

(equivalent to Υ to be a join-semilattice homomor-

phism, if we order functions between small bounded lattices component-
wise).

I call morphisms of such categories

sides

.

1

Remark

2236

.

We could generalize to functions between small join-

semilattices with least elements instead of bounded lattices only, but this is not
really necessary.

Definition

2237

.

I will call objects of the source category of this functor

simply

objects of the presides

.

Definition

2238

.

Bounded

system of presides is system of presides from an

ordered category with bounded Hom-sets such that

X, Y

Ob Src Υ the following

additional axioms hold for all suitable

a

:

1

.

Hom(

X,Y

)

a

=

.

2

.

>

Hom(

X,Y

)

a

=

>

unless

a

=

Definition

2239

.

System of presides with identities

is a system of presides

with a morphism id

a

Src Υ for every object

A

of Src Υ and

a

A

and the

following additional axioms:

1

. id

c

v

1

A

for every

c

A

where

A

is an object of Src Υ.

2

.

h

id

c

i

= (

λx

A

:

x

u

c

) for every

c

A

where

A

is an object of Src Υ

Definition

2240

.

System of sides

is a system of presides which is both bounded

and with identities.

1

The idea for the name is that we consider one “side”

h

f

i

of a funcoid instead of both sides

h

f

i

and

f

1

.