391 research outputs found
Atomic polyadic algebras of infinite dimension are completely representable
We show that atomic polyadic algebras of infinite dimensions are completely
representabl
Finite relation algebras and omitting types in modal fragments of first order logic
Let 2<n\leq l<m< \omega. Let L_n denote first order logic restricted to the
first n variables. We show that the omitting types theorem fails dramatically
for the n--variable fragments of first order logic with respect to clique
guarded semantics, and for its packed n--variable fragments. Both are modal
fragments of L_n. As a sample, we show that if there exists a finite relation
algebra with a so--called strong l--blur, and no m--dimensional relational
basis, then there exists a countable, atomic and complete L_n theory T and type
\Gamma, such that \Gamma is realizable in every so--called m--square model of
T, but any witness isolating \Gamma cannot use less than variables. An
--square model M of T gives a form of clique guarded semantics, where the
parameter m, measures how locally well behaved M is. Every ordinary model is
k--square for any n<k<\omega, but the converse is not true. Any model M is
\omega--square, and the two notions are equivalent if M is countable.
Such relation algebras are shown to exist for certain values of l and m like
for n\leq l<\omega and m=\omega, and for l=n and m\geq n+3. The case l=n and
m=\omega gives that the omitting types theorem fails for L_n with respect to
(usual) Tarskian semantics: There is an atomic countable L_n theory T for which
the single non--principal type consisting of co--atoms cannot be omitted in any
model M of T.
For n<\omega, positive results on omitting types are obained for L_n by
imposing extra conditions on the theories and/or the types omitted. Positive
and negative results on omitting types are obtained for infinitary variants and
extensions of L_{\omega, \omega}.Comment: arXiv admin note: text overlap with arXiv:1408.3282, arXiv:1502.0770
Strongly representable atom structures and neat embeddings
In this paper we give an alternative construction using Monk like algebras
that are binary generated to show that the class of strongly representable atom
structures is not elementary. The atom structures of such algebras are
cylindric basis of relation algebras, both algebras are based on one graph such
that both the relation and cylindric algebras are representable if and only if
the chromatic number of the graph is infinite. We also relate the syntactic
notion of algebras having a (complete) neat embedding property to the
semantical notion of having various forms of (complete) relativized
representations. Finally, we show that for n>5, the problemn as to whether a
finite algebra is in the class SNr_3CA_6 is undecidable. In contrast, we show
that for a finite algebra of arbitary finite dimensions that embed into extra
dimensions of a another finite algebra, then this algebra have a finite
relativized representation. Finally we devise what we call neat games, for such
a game if \pe\ has a \ws \ on an atomic algebra \A in certain atomic game and
\pa has a \ws in another atomic game, then such algebras are elementary
equivalent to neat reducts, but do not have relativized (local) complete
represenations. From such results, we infer that the omitting types theorem for
finite variable fragments fails even if we consider clique guarded semantics.
The size of cliques are determined by the number of pebbles used by \pa\.Comment: arXiv admin note: text overlap with arXiv:1302.1368, arXiv:1304.1149,
arXiv:1305.4570, arXiv:1307.101
An invitation to model theory and C*-algebras
We present an introductory survey to first order logic for metric structures
and its applications to C*-algebras
An Ehrenfeucht-Fra\"{i}ss\'{e} Game for
Ehrenfeucht-Fraisse games are very useful in studying separation and
equivalence results in logic. The standard finite Ehrenfeucht-Fraisse game
characterizes equivalence in first order logic. The standard
Ehrenfeucht-Fraisse game in infinitary logic characterizes equivalence in
. The logic is the extension of first
order logic with countable conjunctions and disjunctions. There was no
Ehrenfeucht-Fraisse game for in the literature.
In this paper we develop an Ehrenfeucht-Fraisse Game for
. This game is based on a game for propositional and first
order logic introduced by Hella and Vaananen. Unlike the standard
Ehrenfeucht-Fraisse games which are modeled solely after the behavior of
quantifiers, this new game also takes into account the behavior of connectives
in logic. We prove the adequacy theorem for this game. We also apply the new
game to prove complexity results about infinite binary strings.Comment: 22 pages, 1 figur
Omitting Types and the Baire Category Theorem
The Omitting Types Theorem in model theory and the Baire Category Theorem in
topology are known to be closely linked. We examine the precise relation
between these two theorems. Working with a general notion of logic we show that
the classical Omitting Types Theorem holds for a logic if a certain associated
topological space has all closed subspaces Baire. We also consider stronger
Baire category conditions, and hence stronger Omitting Types Theorems,
including a game version. We use examples of spaces previously studied in
set-theoretic topology to produce abstract logics showing that the game
Omitting Types statement is consistently not equivalent to the classical one.Comment: 17 page
Expressive power of infinitary [0, 1]-valued logics
We consider model-theoretic properties related to the expressive power of
three analogues of for metric structures. We give an
example showing that one of these infinitary logics is strictly more expressive
than the other two, but also show that all three have the same elementary
equivalence relation for complete separable metric structures. We then prove
that a continuous function on a complete separable metric structure is
automorphism invariant if and only if it is definable in the more expressive
logic. Several of our results are related to the existence of Scott sentences
for complete separable metric structures.Comment: 15 pages, final versio
A polyadic algebra of infinite dimension is completely representable if and only if it is atomic and completely additive
We prove the result in the title. We infer, that unlike cylindric algebras,
there is a first order axiomatization of the class of completely representable
polyadic algebras of infinite dimension, though the one we obtain is infinite;
in fact uncountable, but shares a single schema, stipulating that the
(uncountably many)substitution operators are completely additive. Similar
results are obtained for non commutative reducts of polyadic equality algebras
of infinite dimensions, where we can drop complete additivity. However, it
remains unknown to us whether there are atomic polyadic algebras of infinite
dimension that are not completely additive; but we strongly conjecture that
there are.Comment: Submitted to the Journal of Symbolic Logic. arXiv admin note:
substantial text overlap with arXiv:1301.585
Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras
We study what we call topological cylindric algebras and tense cylindric
algebras defined for every ordinal . The former are cylindric algebras
of dimension expanded with modalities indexed by .
The semantics of representable topological algebras is induced by the interior
operation relative to a topology defined on their bases. Tense cylindric
algebras are cylindric algebras expanded by the modalities (future) and
(past) algebraising predicate temporal logic.
We show for both tense and topological cylindric algebras of finite dimension
that infinitely many varieties containing and including the variety of
representable algebras of dimension are not atom canonical. We show that
any class containing the class of completely representable algebras having a
weak neat embedding property is not elementary. From these two results we draw
the same conclusion on omitting types for finite variable fragments of
predicate topologic and temporal logic. We show that the usual version of the
omitting types theorem restricted to such fragments when the number of
variables is fails dramatically even if we considerably broaden the class
of models permitted to omit a single non principal type in countable atomic
theories, namely, the non-principal type consting of co atoms.Comment: arXiv admin note: substantial text overlap with arXiv:1308.6165,
arXiv:1307.1016, arXiv:1309.0681, arXiv:1307.4298, arXiv:1401.1103,
arXiv:1401.115
A survey on tame abstract elementary classes
Tame abstract elementary classes are a broad nonelementary framework for
model theory that encompasses several examples of interest. In recent years,
progress toward developing a classification theory for them have been made.
Abstract independence relations such as Shelah's good frames have been found to
be key objects. Several new categoricity transfers have been obtained. We
survey these developments using the following result (due to the second author)
as our guiding thread:
If a universal class is categorical in cardinals of arbitrarily high
cofinality, then it is categorical on a tail of cardinals.Comment: 84 page
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