how can we slove cand d Let L be any first order language including equality = a
ID: 3813368 • Letter: H
Question
how can we slove cand d Let L be any first order language including equality = as usual. Recall the definition of isomorphism given in our text. a) Let A, B, C denote any L-structures and let approximately be any isomorphism between L structures. Show the following claims are true: (a) A approximately A. (b) If A approximately B then B approximately A. (c) If A approximately B and B approximately C then A approximately C. b) Deduce that approximately defines an equivalence relation on the set of all L-structures. This equivalence relation created by approximately immediately gives rise to a partition of the set of all models suitable for L. Explain how to define such a partition using approximately. c) Prove there are infinitely many, equivalence classes of approximately for any language FOL L. d) Recoil our notion of two models A, B being elementary equivalent, denoted A B. This relation models also defines an equivalence relation on L-structures. Explain how these two equivalence relations on the space of L-models relate to one another.Explanation / Answer
A countable Borel equivalence relation on a standard Borel space X is a Borel equivalence relation E X2 with the
property that every equivalence class [x]E, x X, is countable. Wedenote by E the class of countable Borel equivalence
relations. Over the last 25 years there has been an extensive study of countable Borel equivalence relations and their
connection with group actions and ergodic theory. An important aspect of this work is an understanding of the kind of
countable (first-order) structures that can be assigned in a uniform Borel way to each class of a
given equivalence relation. This is made precise in the following definitions; see [JKL], Section 2.5.
Let L = {Ri
| i I} be a countable relational language, where Ri has arity ni
, and K a class
of countable structures in L closed under isomorphism. Let E be a countable Borel equivalence
relation on a standard Borel space X. An L-structure on E is a Borel structure A = (X, RA
i
)iI
of L with universe X (i.e., each RA
i Xni
is Borel) such that for i I and x1, . . . , xni X,
RA
i
(x1, . . . , xni
) = x1 E x2 E · · · E xni
. Then each E-class C is the universe of the countable
L-structure A|C. If for all such C, A|C K, we say that A is a K-structure on E. Finally if E
admits a K-structure, we say that E is K-structurable.
Many important classes of countable Borel equivalence relations can be described as the Kstructurable
relations for appropriate K. For example, the hyperfinite equivalence relations are
exactly the K-structurable relations, where K is the class of linear orderings embeddable in Z. The
treeable equivalence relations are the K-structurable relations, where K is the class of countable
trees (connected acyclic graphs). The equivalence relations generated by a free Borel action of a
countable group are the K-structurable relations, where K is the class of structures corresponding
to free transitive -actions. The equivalence relations admitting no invariant probability Borel
measure are the K-structurable relations, where L = {R, S}, R unary and S binary, and K consists
of all countably infinite structures A = (A, RA, SA), with RA an infinite, co-infinite subset of A and
S
A the graph of a bijection between A and RA.
For L = {Ri
| i I} as before and countable set X, we denote by ModX(L) the standard Borel
space of countable L-structures with universe X. Clearly every countable L-structure is isomorphic
to some A ModX(L), for X {1, 2, . . . , N}. Given a class K of countable L-structures, closed
under isomorphism, we say that K is Borel if KModX(L) is Borel in ModX(L), for each countable
set X. We are interested in Borel classes K in this paper. For any L1-sentence , the class of
countable models of is Borel. By a classical theorem of Lopez-Escobar [LE], every Borel class K
of L-structures is of this form, for some L1-sentence . We will also refer to such as a theory.
Research partially supported by NSERC PGS D
†Research partially supported by NSF Grants DMS-0968710 and DMS-1464475
1
Adopting this model-theoretic point of view, given a theory and a countable Borel equivalence
relation E, we put
E |=
if E is K-structurable, where K is the class of countable models of , and we say that E is -
structurable if E |= . We denote by E E the class of -structurable countable Borel equivalence
relations. Finally we say that a class C of countable Borel equivalence relations is elementary
if it is of the form E, for some (which axiomatizes C). In some sense the main goal of this
paper is to study the global structure of elementary classes.
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