Some remarks concerning the infinite distributivity of lattices

1970 ◽  
Vol 8 (1) ◽  
pp. 523-527
Author(s):  
M. G. Rabinovich
Keyword(s):  
Infinite Distributivity

Infinite Distributivity in the Lattice of Varieties of m-groups

2015 ◽  
Vol 54 (1) ◽  
pp. 1-9
Author(s):  
N. V. Bayanova ◽  
A. V. Zenkov
Keyword(s):  
Lattice Of Varieties ◽  
Infinite Distributivity

A lattice of interpretability types of theories

1977 ◽  
Vol 42 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Order Theory ◽  
Partial Ordering ◽  
First Order ◽  
First Order Theory ◽  
Relative Consistency ◽  
Infinite Distributivity ◽  
Closed Fields ◽  
Image Position ◽  
Interpretability Type ◽  
Consistency Proofs

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

Sequential theories and infinite distributivity in the lattice of chapters

1989 ◽  
Vol 54 (1) ◽  
pp. 190-206 ◽  
Author(s):  
Alan S. Stern
Keyword(s):  
Elementary Proof ◽  
Infinite Distributivity ◽  
Distributive Law ◽  
Image Position

AbstractWe introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive lawin the lattice of chapters, in which the chapters a and bi are compact. (Counterexamples in which a is not compact have been found previously.)

Infinite distributivity in certain function lattices

1978 ◽  
Vol 31 (1) ◽  
pp. 182-184
Author(s):  
J�rg Schmid
Keyword(s):  
Infinite Distributivity

Formations of lattice ordered groups and of GMV-algebras

2008 ◽  
Vol 58 (5) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Infinite Distributivity ◽  
Subdirect Products

AbstractA class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ1 and ℱ2 the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ1 and ℱ2 are partially ordered by the class-theoretical inclusion. We prove that ℱ1 satisfies the infinite distributivity law

scholarly journals Imbedding Infinitely Distributive Lattices Completely Isomorphically Into Boolean Algebras

1959 ◽  
Vol 15 ◽  
pp. 71-81 ◽  
Author(s):  
Nenosuke Funayama
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Algebras ◽  
The Other ◽  
Distributive Lattices ◽  
Algebra 1 ◽  
Infinite Distributivity ◽  
Distributive Law ◽  
Finite Sums ◽  
Finite Products

It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).

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