数学论文（二）_数学联邦政治世界观

3. Special classes of Boolean algebras

There are many special classes of Boolean algebra which are important both for the intrinsic theory of BAs and for applications:

Atomic BAs, already mentioned above.

Atomless BAs, which are defined to be BAs without any atoms. For example, any infinite free BA is atomless.

Complete BAs, defined above. These are specially important in the foundations of set theory.

Interval algebras. These are derived from linearly ordered sets (L,＜) with a first element as follows. One takes the smallest algebra of subsets of L containing all of the half-open intervals [a,b) with a in L and b in L or equal to ∞. These BAs are useful in the study of Lindenbaum-Tarski algebras. Every countable BA is isomorphic to an interval algebra, and thus a countable BA can be described by indicating an ordered set such that it is isomorphic to the corresponding interval algebra.

Tree algebras. A tree is a partially ordered set (T,＜) in which the set of predecessors of any element is well-ordered. Given such a tree, one considers the algebra of subsets of T generated by all sets of the form {b:a≤b} for some a in T.

Superatomic BAs. These are BAs which are not only atomic, but are such that each subalgebra and homomorphic image is atomic.

4. Structure theory and cardinal functions on Boolean algebras

Much of the deeper theory of Boolean algebras, telling about their structure and classification, can be formulated in terms of certain functions defined for all Boolean algebras, with infinite cardinals as values. We define some of the more important of these cardinal functions, and state some of the known structural facts, mostly formulated in terms of them

The cellularity c(A) of a BA is the supremum of the cardinalities of sets of pairwise disjoint elements of A.

A subset X of a BA A is independent if X is a set of free generators of the subalgebra that it generates. The independence of A is the supremum of cardinalities of independent subsets of A.

A subset X of a BA A is dense in A if every nonzero element of A is ≥ a nonzero element of X. The π-weight of A is the smallest cardinality of a dense subset of A.

Two elements x,y of A are incomparable if neither one is ≤ the other. The supremum of cardinalities of subset X of A consisting of pairwise incomparable elements is the incomparability of A.

A subset X of A is irredundant if no element of X is in the subalgebra generated by the others.

An important fact concerning cellularity is the Erdős-Tarski theorem: if the cellularity of a BA is a singular cardinal, then there actually is a set of disjoint elements of that size; for cellularity regular limit (inaccessible), there are counterexamples. Every infinite complete BA has an independent subset of the same size as the algebra. Every infinite BA A has an irredundant incomparable subset whose size is the π-weight of A. Every interval algebra has countable independence. A superatomic algebra does not even have an infinite independent subset. Every tree algebra can be embedded in an interval algebra. A BA with only the identity automorphism is called rigid. There exist rigid complete BAs, also rigid interval algebras and rigid tree algebras.

More recently, many cardinal functions of min-max type have been studied. For example, small independence is the smallest size of an infinite maximal independent set; and small cellularity is the smallest size of an infinite partition of unity.

5. Decidability and undecidability questions

A basic result of Tarski is that the elementary theory of Boolean algebras is decidable. Even the theory of Boolean algebras with a distinguished ideal is decidable. On the other hand, the theory of a Boolean algebra with a distinguished subalgebra is undecidable. Both the decidability results and undecidablity results extend in various ways to Boolean algebras in extensions of first-order logic.

6. Lindenbaum-Tarski algebras

A very important construction, which carries over to many logics and many algebras other than Boolean algebras, is the construction of a Boolean algebra associated with the sentences in some logic. The simplest case is sentential logic. Here there are sentence symbols, and common connectives building up longer sentences from them: disjunction, conjunction, and negation. Given a set A of sentences in this language, two sentences s and t are equivalent modulo A if and only if the biconditional between them is a logical consequence of A. The equivalence classes can be made into a BA such that + corresponds to disjunction, ⋅ to conjunction, and − to negation. Any BA is isomorphic to one of this form. One can do something similar for a first-order theory. Let T be a first-order theory in a first-order language L. We call formulas ϕ and ψ equivalent provided that T⊢ϕ↔ψ. The equivalence class of a sentence ϕ is denoted by [ϕ]. Let A be the collection of all equivalence classes under this equivalence relation. We can make A into a BA by the following definitions, which are easily justified:

[ϕ]+[ψ]=[ϕ∨ψ]

[ϕ]⋅[ψ]=[ϕ∧ψ]

−[ϕ]=[¬ϕ]

0=[F]

1=[T]

Every BA is isomorphic to a Lindenbaum-Tarski algebra. However, one of the most important uses of these classical Lindenbaum-Tarski algebras is to describe them for important theories (usually decidable theories). For countable languages this can be done by describing their isomorphic interval algebras. Generally this gives a thorough knowledge of the theory. Some examples are:

TheoryIsomorphic to interval algebra on

(1)essentially undecidable theoryQ, the rationals

(2)BAsN×N, square of the positive integers, ordered lexicographically

(3)linear ordersA×Q ordered antilexicographically, where A is NN in its usual order

(4)abelian groups(Q+A)×Q

7. Boolean-valued models

In model theory, one can take values in any complete BA rather than the two-element BA. This Boolean-valued model theory was developed around 1950–1970, but has not been worked on much since. But a special case, Boolean-valued models for set theory, is very much at the forefront of current research in set theory. It actually forms an equivalent way of looking at the forcing construction of Cohen, and has some technical advantages and disadvantages. Philosophically it seems more satisfactory than the forcing concept. We describe this set theory case here; it will then become evident why only complete BAs are considered. Let B be a complete BA. First we define the Boolean valued universe V(B). The ordinary set-theoretic universe can be identified with V(2), where 2 is the 2-element BA. The definition is by transfinite recursion, where α,β are ordinals and λ is a limit ordinal:

V(B,0)=∅

V(B,α+1)={f:dom(f)⊂V(B,α) and range(f)⊂B}

V(B,λ)=

⋃

β＜λ

V(B,β).

where dom(f) is the domain of function f and range(f) is the range of function f. The B-valued universe is the proper class V(B) which is the union of all of these Vs. Next, one defines by a rather complicated transfinite recursion over well-founded sets the value of a set-theoretic formula with elements of the Boolean valued universe assigned to its free variables:

‖x∈y‖=

∑

t∈dom(y)

‖x=t‖⋅y(t)

‖x⊆y‖=

∏

t∈dom(x)

−x(t)+‖t∈y‖

‖x=y‖=‖x⊆y‖⋅‖y⊆x‖

‖¬ϕ‖=−‖ϕ‖

‖ϕ∨ψ‖=‖ϕ‖+‖ψ‖

‖∃xϕ(x)‖=