On initial segments of hyperdegrees

1970 ◽  
Vol 35 (2) ◽  
pp. 189-197 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Finite Distributive Lattice ◽  
Order Types

An initial segment of hyperdegrees is a set S of hyperdegrees such that whenever h ∈ S and k ≦ h then k ∈ S. The main results of this paper affirm the existence of initial segments having certain order types. In particular, if L is a finite distributive lattice then L is isomorphic to an initial segment of hyperdegrees [Theorem 1]; as a consequence the elementary theory of the ordering of hyperdegrees is recursively undecidable [Corollary 1].

Noninitial segments of the α-degrees

1973 ◽  
Vol 38 (3) ◽  
pp. 368-388 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Recursion Theory ◽  
Partial Ordering ◽  
First Order ◽  
Order Language ◽  
Suitable Notion ◽  
The Universe ◽  
Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

scholarly journals Congruence Lattices of Finite Semimodular Lattices

1998 ◽  
Vol 41 (3) ◽  
pp. 290-297 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser ◽  
E. T. Schmidt
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Finite Distributive Lattice ◽  
Congruence Lattices ◽  
Semimodular Lattices

AbstractWe prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

scholarly journals The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

scholarly journals Every finite distributive lattice is a set of stable matchings for a small stable marriage instance

1987 ◽  
Vol 44 (2) ◽  
pp. 304-309 ◽  
Author(s):  
Dan Gusfield ◽  
Robert Irving ◽  
Paul Leather ◽  
Michael Saks
Keyword(s):  
Distributive Lattice ◽  
Stable Marriage ◽  
Finite Distributive Lattice ◽  
Stable Matchings

scholarly journals Every finite distributive lattice is a set of stable matchings

1984 ◽  
Vol 37 (3) ◽  
pp. 353-356 ◽  
Author(s):  
Charles Blair
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice ◽  
Stable Matchings

Congrunce lattices of rectangular lattices1

2020 ◽  
Vol 39 (3) ◽  
pp. 2831-2843
Author(s):  
Peng He ◽  
Xue-Ping Wang
Keyword(s):  
Distributive Lattice ◽  
Upper Bound ◽  
Congruence Lattice ◽  
Semimodular Lattice ◽  
Rectangular Lattice ◽  
Finite Distributive Lattice ◽  
Irreducible Elements

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.

scholarly journals On the Strong Purity of the Sublattice-Lattice of a Finite Distributive Lattice

1983 ◽  
Vol 06 (2) ◽  
pp. 381-388
Author(s):  
C. C. CHEN ◽  
K. M. KOH
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice

On the purity of the lattice of sublattices of a finite distributive lattice

1982 ◽  
Vol 15 (1) ◽  
pp. 258-271
Author(s):  
C. C. Chen ◽  
K. M. Koh ◽  
S. C. Lee
Keyword(s):  
Distributive Lattice ◽  
Finite Distributive Lattice

On the T-degrees of partial functions

1985 ◽  
Vol 50 (3) ◽  
pp. 580-588 ◽  
Author(s):  
Paolo Casalegno
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Order Theory ◽  
Partial Functions ◽  
First Order ◽  
First Order Theory ◽  
Degrees Of Unsolvability

AbstractLet 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

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