Homotopy and Homology of Finite Lattices

10.37236/1723 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Simplicial Complex ◽  
Finite Lattice ◽  
Order Complex ◽  
Homotopy Equivalence ◽  
Finite Lattices ◽  
Geometric Lattices

We exhibit an explicit homotopy equivalence between the geometric realizations of the order complex of a finite lattice and the simplicial complex of coreless sets of atoms whose join is not ${}\hat 1{}$. This result, which extends a theorem of Segev, leads to a description of the homology of a finite lattice, extending a result of Björner for geometric lattices.

scholarly journals The Canonical Join Complex

10.37236/7866 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Emily Barnard
Keyword(s):  
Number Theory ◽  
Simplicial Complex ◽  
Finite Lattice ◽  
Noncrossing Partitions ◽  
Prime Factorization ◽  
Finite Lattices ◽  
Abstract Simplicial Complex ◽  
Order Ideals ◽  
Minimal Factorization

A canonical join representation is a certain minimal "factorization" of an element in a finite lattice $L$ analogous to the prime factorization of an integer from number theory. The expression $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where "lowest" is made precise by comparing order ideals under containment). Canonical join representations appear in many familiar guises, with connections to comparability graphs and noncrossing partitions. When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

On finite lattices of degrees of constructibility of reals

1976 ◽  
Vol 41 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Standard Model ◽  
Finite Lattice ◽  
Partial Ordering ◽  
Turing Degrees ◽  
Upper Semilattice ◽  
Finite Lattices ◽  
Definition Of ◽  
The Universe

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ωi such that the following holds:(1) For any s, s′ Є Ui, i Є ω, k, m Є l, k ≤Km & s(m) = s′(m) → s(k) = s′(k).(2) For any s Є Ui, i Є ω, s(0) = 0 where 0 is the least element of K.(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where vK denotes the join in K.

Improved finite-lattice method for estimating the zero-temperature properties of two-dimensional lattice models

1996 ◽  
Vol 74 (1-2) ◽  
pp. 54-64 ◽  
Author(s):  
D. D. Betts ◽  
S. Masui ◽  
N. Vats ◽  
G. E. Stewart
Keyword(s):  
Spontaneous Magnetization ◽  
Finite Lattice ◽  
Quantum Spin ◽  
Square Lattice ◽  
Two Dimensional ◽  
Zero Temperature ◽  
Quantum Spin Systems ◽  
Lattice Method ◽  
Dimensional Lattice ◽  
Finite Lattices

The well-known finite-lattice method for the calculation of the properties of quantum spin systems on a two-dimensional lattice at zero temperature was introduced in 1978. The method has now been greatly improved for the square lattice by including finite lattices based on parallelogram tiles as well as the familiar finite lattices based on square tiles. Dozens of these new finite lattices have been tested and graded using the [Formula: see text] ferromagnet. In the process new and improved estimates have been obtained for the XY model's ground-state energy per spin, ε0 = −0.549 36(30) and spontaneous magnetization per spin, m = 0.4349(10). Other properties such as near-neighbour, zero-temperature spin–spin correlations, which appear not to have been calculated previously, have been estimated to high precision. Applications of the improved finite-lattice method to other models can readily be carried out.

On finite lattices of degrees of constructibility

1977 ◽  
Vol 42 (3) ◽  
pp. 349-371 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Finite Lattice ◽  
Partial Ordering ◽  
Generic Model ◽  
Crucial Point ◽  
Generic Models ◽  
Ordinal Numbers ◽  
Finite Lattices ◽  
Definition Of ◽  
Two Stages ◽  
The Given

We shall prove the following theorem:Theorem. For any finite lattice there is a model of ZF in which the partial ordering of the degrees of constructibility is isomorphic with the given lattice.Let M be a standard countable model of ZF satisfying V = L. Let K be the given finite lattice. We shall extend M by forcing.The paper is divided into two parts. The first part concerns the definition of the set of forcing conditions and some properties of this set expressible without the use of generic filters.We define first a representation of a lattice and then the set of conditions. In Lemmas 1, 2 we show that there are some canonical isomorphisms between some conditions and that a single condition has some canonical automorphisms.In Lemma 3 and Definition 7 we show some methods of defining conditions. We shall use those methods in the second part to define certain conditions with special properties.Lemma 4 gives a connection between the sets P and Pk (see Definitions 4 and 5). It is next employed in the second part in Lemma 10 in an essential way.Indeed, Lemma 10 is necessary for Lemma 13, which is the crucial point of the whole construction. Lemma 5 is also employed in Lemma 13 (exactly in its Corollary).The second part of the paper is devoted to the examination of the structure of degrees of constructibility in a generic model. First, we show that degrees of some “sections” of a generic real (Definition 9) form a lattice isomorphic with K. Secondly, we show that there are no other degrees in the generic model; this is the most difficult property to obtain by forcing. We prove, in two stages, that it holds in our generic models. We first show, by using special properties of the forcing conditions, that sets of ordinal numbers have no other degrees. Then we show that the degrees of sets of ordinals already determine the degrees of other sets.

Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices

1979 ◽  
Vol 31 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Alan Day
Keyword(s):  
Lattice Theory ◽  
Finite Lattice ◽  
Free Lattice ◽  
Finite Lattices

In [8], McKenzie introduced the notion of a bounded homomorphism between lattices, and, using this concept, proved several deep results in lattice theory. Some of these results were intimately connected with the work of Jónsson and Kiefer in [6] where an attempt was made to characterize finite sublattices of free lattices. McKenzie's characterization and others that followed (see [7] and [5]) still have not answered the (now) celebrated Jônsson conjecture:A finite lattice is a sublattice of a free lattice if and only if it satisfies (SD∨), (SD∨) and (W).(The properties mentioned here are defined in the text.)

Finite Lattices of Projections in Factors and Approximately Finite C*-Algebras

1992 ◽  
Vol 35 (1) ◽  
pp. 116-125
Author(s):  
S. C. Power
Keyword(s):  
Tensor Product ◽  
General Result ◽  
Finite Lattice ◽  
Tensor Products ◽  
C Algebras ◽  
Finite Lattices

AbstractA unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

RANDOM WALKS, SELF-AVOIDING WALKS AND 1/f NOISE ON FINITE LATTICES

2004 ◽  
Vol 04 (03) ◽  
pp. L413-L424 ◽  
Author(s):  
FERDINAND GRÜNEIS
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Lattice Structure ◽  
Finite Lattice ◽  
The Self ◽  
Random Walker ◽  
Finite Lattices ◽  
Self Avoiding Walks

We investigate the probabilities for a return to the origin at step n of a random walker on a finite lattice. As a consistent measure only the first returns to the origin appear to be of relevance; these include paths with self-intersections and self-avoiding polygons. Their return probabilities are power-law distributed giving rise to 1/f b noise. Most striking is the behavior of the self-avoiding polygons exhibiting a slope b=0.83 for d=2 and b=0.93 for d=3 independent on lattice structure.

Lattice embeddings into the recursively enumerable degrees. II

1989 ◽  
Vol 54 (3) ◽  
pp. 735-760 ◽  
Author(s):  
K. Ambos-Spies ◽  
M. Lerman
Keyword(s):  
Decision Procedure ◽  
Finite Lattice ◽  
Labeled Trees ◽  
Recursively Enumerable ◽  
Finite Lattices ◽  
Embeddability Condition

The problem of characterizing the finite lattices which can be embedded into the recursively enumerable degrees has a long history, which is summarized in [AL]. This problem is an important one, as its solution is necessary if a decision procedure for the ∀∃-theory of the poset of recursively emumerable degrees is to be found. A recursive nonembeddability condition, NEC, which subsumes all known nonembeddability conditions was presented in [AL]. This paper focuses on embeddability. An embeddability condition, EC, is introduced, and we prove that every finite lattice having EC can be embedded (as a lattice) into . EC subsumes all known embeddability conditions.EC is a Π3 condition which states that certain obstructions to proving embeddability do not exist. It seems likely that the recursive labeled trees used in EC can be replaced with trees which are effectively generated from uniformly defined finite trees, in which case EC would be equivalent to a recursive condition. We do not know whether EC and NEC are complementary. This problem seems to be combinatorial, rather than recursion-theoretic in nature. Our efforts to find a finite lattice satisfying neither EC nor NEC have, to this point, been unsuccessful. It is the second author's conjecture that the techniques for proving embeddability which are used in this paper cannot be refined very much to obtain new embeddability results.EC is introduced in §2, and the various conditions and definitions are motivated by presenting examples of embeddable lattices and indicating how the embedding proof works in those particular cases. The embedding construction is presented in §3, and the proof in §4.

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