130 From Classical Mechanics to Quantum Field Theory. A Tutorial
Definition 2.3.5.IfX,Y are lattices, a maph:X→Y is a(lattice) homo-
morphismwhen
h(a∨Xb)=h(a)∨Yh(b),h(a∧Xb)=h(a)∧Yh(b),a,b∈X(with the obvious notations.) IfX andY are bounded, a homomorphismhis
further required to satisfy
h( (^0) X)= (^0) Y,h( (^1) X)= (^1) Y.
IfXandYare orthocomplemented, a homomorphismhalso satisfies
h(¬Xa)=¬Yh(x).
IfX,Yareσ-complete,hfurther fulfills
h(∨n∈Nan)=∨n∈Nh(an),if{an}n∈N⊂X.
In all cases (bounded, orthocomplemented, σ-complete lattices, Boolean
(σ-)algebras) ifhis bijective it is calledisomorphismof the relative structures.
It is clear that, just because it is a concreteσ-algebra, the lattice of the elementary
properties of a classical system is a lattice which isdistributive,bounded(here 0 =
∅and1=Γ),orthocomplemented(the orthocomplement being the complement
with respect to Γ) andσ-complete. Moreover, as the reader can easily prove, the
above map,B(R)B→EB(f), is also a homomorphism of Booleanσ-algebras.
Remark 2.3.6.Given an abstract Booleanσ-algebraX, does there exist a concrete
σ-algebra of sets that is isomorphic to theprevious one? In this respect the fol-
lowing general result holds, known asLoomis-Sikorski theorem.^12 This guarantees
that every Booleanσ-algebra is isomorphic to a quotient Booleanσ-algebraΣ/N,
whereΣis a concreteσ-algebra of sets over a measurable space andN⊂Σis closed
under countable unions; moreover,∅∈Nand for anyA∈ΣwithA⊂N∈N,
thenA∈N. The equivalence relation isA∼BiffA∪B(A∩B)∈N, for any
A, B∈Σ. It is easy to see the coset spaceΣ/Ninherits the structure of Boolean
σ-algebra fromΣwith respect to the (well-defined) partial order relation[A]≥[B]
ifA⊃B,A, B∈Σ.
(^12) Sikorski S.: On the representation of Boolean algebras as field of sets. Fund. Math. 35 ,
247-256 (1948).