453 research outputs found
A generalization of Goodstein's theorem: interpolation by polynomial functions of distributive lattices
We consider the problem of interpolating functions partially defined over a
distributive lattice, by means of lattice polynomial functions. Goodstein's
theorem solves a particular instance of this interpolation problem on a
distributive lattice L with least and greatest elements 0 and 1, resp.: Given
an n-ary partial function f over L, defined on all 0-1 tuples, f can be
extended to a lattice polynomial function p over L if and only if f is
monotone; in this case, the interpolating polynomial p is unique. We extend
Goodstein's theorem to a wider class of n-ary partial functions f over a
distributive lattice L, not necessarily bounded, where the domain of f is a
cuboid of the form D={a1,b1}x...x{an,bn} with ai<bi, and determine the class of
such partial functions which can be interpolated by lattice polynomial
functions. In this wider setting, interpolating polynomials are not necessarily
unique; we provide explicit descriptions of all possible lattice polynomial
functions which interpolate these partial functions, when such an interpolation
is available.Comment: 12 page
On centralizers of finite lattices and semilattices
We study centralizer clones of finite lattices and semilattices. For
semilattices, we give two characterizations of the centralizer and also derive
formulas for the number of operations of a given essential arity in the
centralizer. We also characterize operations in the centralizer clone of a
distributive lattice, and we prove that the essential arity of operations in
the centralizer is bounded for every finite (possibly nondistributive) lattice.
Using these results, we present a simple derivation for the centralizers of
clones of Boolean functions.Comment: 21 pages, 4 figure
Orthomodular-Valued Models for Quantum Set Theory
In 1981, Takeuti introduced quantum set theory by constructing a model of set
theory based on quantum logic represented by the lattice of closed linear
subspaces of a Hilbert space in a manner analogous to Boolean-valued models of
set theory, and showed that appropriate counterparts of the axioms of
Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model.
In this paper, we aim at unifying Takeuti's model with Boolean-valued models by
constructing models based on general complete orthomodular lattices, and
generalizing the transfer principle in Boolean-valued models, which asserts
that every theorem in ZFC set theory holds in the models, to a general form
holding in every orthomodular-valued model. One of the central problems in this
program is the well-known arbitrariness in choosing a binary operation for
implication. To clarify what properties are required to obtain the generalized
transfer principle, we introduce a class of binary operations extending the
implication on Boolean logic, called generalized implications, including even
non-polynomially definable operations. We study the properties of those
operations in detail and show that all of them admit the generalized transfer
principle. Moreover, we determine all the polynomially definable operations for
which the generalized transfer principle holds. This result allows us to
abandon the Sasaki arrow originally assumed for Takeuti's model and leads to a
much more flexible approach to quantum set theory.Comment: 25 pages, v2: to appear in Rev. Symb. Logic, v3: corrected typo
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