Monotone Galois connections Galois connection

1 monotone galois connections

1.1 power set; implication , conjunction
1.2 lattices
1.3 transitive group actions
1.4 image , inverse image
1.5 span , closure
1.6 syntax , semantics

monotone galois connections
power set; implication , conjunction

for order theoretic example, let u set, , let , b both power set of u, ordered inclusion. pick fixed subset l of u. maps f , g, f(m) = l ∩ m, , g(n) = n ∪ (u \ l), form monotone galois connection, f being lower adjoint. similar galois connection lower adjoint given meet (infimum) operation can found in heyting algebra. especially, present in boolean algebra, 2 mappings can described f(x) = (a ∧ x) , g(y) = (y ∨ ¬a) = (a ⇒ y). in logical terms: implication upper adjoint of conjunction .

lattices

further interesting examples galois connections described in article on completeness properties. speaking, turns out usual functions ∨ , ∧ lower , upper adjoints diagonal map x → x × x. least , greatest elements of partial order given lower , upper adjoints unique function x → {1}. going further, complete lattices can characterized existence of suitable adjoints. these considerations give impression of ubiquity of galois connections in order theory.

transitive group actions

let g act transitively on x , pick point x in x. consider

b


=
{
b
⊆
x
:
x
∈
b
;
∀
g
∈
g
,
g
b
=
b
 

o
r

 
g
b
∩
b
=
∅
}
,


{\displaystyle {\mathcal {b}}=\{b\subseteq x:x\in b;\forall g\in g,gb=b\ \mathrm {or} \ gb\cap b=\emptyset \},}

the set of blocks containing x. further, let





g




{\displaystyle {\mathcal {g}}}

consist of subgroups of g containing stabilizer of x.

then, correspondence





b


→


g




{\displaystyle {\mathcal {b}}\to {\mathcal {g}}}

:

b
↦

h

b


=
{
g
∈
g
:
g
x
∈
b
}


{\displaystyle b\mapsto h_{b}=\{g\in g:gx\in b\}}

is monotone, one-to-one galois connection. corollary, 1 can establish doubly transitive actions have no blocks other trivial ones (singletons or whole of x): follows stabilizers being maximal in g in case. see doubly transitive group further discussion.

image , inverse image

if  f  : x → y function, subset m of x can form image f(m) =  f (m) = {f(m) | m ∈ m} , subset n of y can form inverse image g(n) =  f (n) = {x ∈ x |  f (x) ∈ n}. f , g form monotone galois connection between power set of x , power set of y, both ordered inclusion ⊆. there further adjoint pair in situation: subset m of x, define h(m) = {y ∈ y |  f ({y}) ⊆ m}. g , h form monotone galois connection between power set of y , power set of x. in first galois connection, g upper adjoint, while in second galois connection serves lower adjoint.

in case of quotient map between algebraic objects (such groups), connection called lattice theorem: subgroups of g connect subgroups of g/n, , closure operator on subgroups of g given h = hn.

span , closure

pick mathematical object x has underlying set, instance group, ring, vector space, etc. subset s of x, let f(s) smallest subobject of x contains s, i.e. subgroup, subring or subspace generated s. subobject u of x, let g(u) underlying set of u. (we can take x topological space, let f(s) closure of s, , take subobjects of x closed subsets of x.) f , g form monotone galois connection between subsets of x , subobjects of x, if both ordered inclusion. f lower adjoint.

syntax , semantics

a general comment of william lawvere syntax , semantics adjoint: take set of logical theories (axiomatizations), , b power set of set of mathematical structures. theory t ∈ a, let f(t) set of structures satisfy axioms t; set of mathematical structures s, let g(s) minimum of axiomatizations approximate s. can f(t) subset of s if , if t logically implies g(s): semantics functor f , syntax functor g form monotone galois connection, semantics being lower adjoint.