Nondistributive Lattices Research Articles

Lattice embeddings into the recursively enumerable degrees

The classification of algebraic structures which can be embedded into ℛ, the uppersemilattice of recursively enumerable degrees, is the key to answering certain questions about Th(ℛ), the elementary theory of ℛ. In particular, these classification problems are important for answering decidability questions about fragments of Th(ℛ). Thus the solutions of Fried berg [F] and Mučnik [M] to Post's problem were easily extended to show that all finite partially ordered sets are embeddable into ℛ, and hence that ∃1 ∩ Th(ℛ), the existential theory of ℛ, is decidable. (The language used is ℒ′, the pure predicate calculus together with a binary relation symbol ≤ to be interpreted as the ordering of ℛ) The problem of determining which finite lattices are embeddable into ℛ has been a long-standing open problem, and is one of the major obstacles to determining whether ∀2 ∩ Th(ℛ), the universal-existential theory of ℛ, is decidable. Shore has obtained some nice partial results in this direction. Embeddings also played a central role in showing that Th(ℛ) is not ℵ0-categorical (Lerman, Shore and Soare [LeShSo]), thus resolving a problem posed by Jockusch. Harrington and Shelah [HS] embedded all 0′-presentable partially ordered sets into ℛ in such a way that the partially ordered sets can be uniformly recovered from four parameters. They used these embeddings to show that Th(ℛ) is undecidable.The first nontrivial extension of the embeddings of Friedberg and Mučnik to lattice embeddings was obtained independently by Lachlan [La1] and Yates [Y] who showed that the four-element Boolean algebra can be embedded into ℛ. Thomason [T] and Lerman independently extended this result to include all finite distributive lattices. The nondistributive case, however, was much more difficult. Lachlan [La2] embedded the two five-element nondistributive lattices M5 and N5 (see Figures 1 and 2) into ℛ, and his proof could easily have been extended to include a larger class of lattices.

Definable principal congruences in congruence distributive varieties

We present an algorithm that, given a finite algebraA generating a congruence distributive (CD) variety, determines whether this variety has first order definable principal congruences (DPC). In fact, DPC turns out to be equivalent to the extendability of the principal congruences of certain subalgebras of the algebras in HS(A 3). To verify this algorithm, we investigate combinatorial properties of finite subdirect powers ofA. Our theorem has a relatively simple formulation for arithmetical algebras. As an application, we obtain McKenzie's result that there are no nondistributive lattice varieties with DPC.

Embedding the diamond lattice in the recursively enumerable truth-table degrees

It is shown that the four element Boolean algebra can be embedded in the recursively enumerable truth-table degrees with least and greatest elements preserved. Corresponding results for other lattices and other reducibilites are also discussed. For sets A, B c w, we say that A is a truth-table (tt) reducible to B if there exists an effective procedure for reducing any question of the form n E A? to an equivalent finite Boolean combination of questions of the form k E B? Then, A, B are said to have the same tt-degree if each is tt-reducible to the other, and tt-degrees have a natural ordering induced by tt-reducibility. (See [1, 6 and 8] for information on tt-degrees.) We show the existence of two incomparable recursively enumerable (r.e.) tt-degrees with supremum 0' (the highest r.e. tt-degree) and infimum 0 (the lowest). In other words, the four-element Boolean algebra (known also as the diamond lattice) can be embedded as a lattice in the r.e. tt-degrees with least and greatest elements preserved. We also obtain analogous results with the diamond lattice replaced by each of the two five-element nondistributive lattices (pentagon and 1-3-1) and with tt-reducibility replaced by many of its restricted forms, such as bounded truth-table and positive reducibility [2]. The history of this problem is as follows. A. H. Lachlan proved in his well-known nondiamond theorem [5, Theorem 5] that the diamond lattice cannot be embedded in the r.e. Turing degrees with 0 and 1 preserved. His proof simultaneously establishes the corresponding result for r.e. weak truth-table (wtt) degrees [6]. Lachlan also showed in [4] that no two incomparable r.e. many-one (m) degrees can have supremum 0', so the diamond lattice cannot be embedded in the r.e. m-degrees with 1 preserved. The trend of these results makes it reasonable to conjecture that the diamond lattice cannot be embedded in the r.e. tt-degrees with 0 and 1 preserved, although in the other direction D. Posner [7] proved that the Turing degrees below 0' are complemented. In [6, Theorem 6.6] P. G. Odifreddi announced that in fact the diamond lattice can be embedded in the r.e. tt-degrees with 0 and 1 preserved. His construction involved splitting a creative set K into two disjoint r.e. Received by the editors April 9, 1984. 1980 Mathematics Subject Classification. Primary 03D30; Secondary 03D25.

Join-prime elements in semidistributive lattices

In this paper we shall discuss some aspects of the relation between the structure of a finite semidistributive lattice L and the set P(L) of its join-prime elements. These investigations were motivated in part by the results of the first author in [2] and [3], wherein finite sublattices of a free lattice which are generated by both their join-prime and meet-prime elements are described. It is well-known that the set of join-prime elements of a finite distributive lattice (which coincides with the set of join-irreducible elements) completely determines the lattice. On the other hand, a finite nondistributive lattice (e.g., M3) may well "have P(L) empty. In a nontrivial finite semidistributive lattice, though, P(L) is never empty (see w and in the presence of Whitman's condition or L being a bounded homomorphic image of a free lattice, we can establish at least some explicit connection between P(L) and the structure of L (w and w For finite lattices in general, we make the following simple observation.

Quantum mechanics of space and time

A formulation of relativistic quantum mechanics is presented independent of the theory of Hilbert space and also independent of the hypothesis of spacetime manifold. A hierarchy is established in the nondistributive lattice of physical ensembles, and it is shown that the projections relating different members of the hierarchy form a semigroup. It is shown how to develop a statistical theory based on the definition of a statistical operator. Involutions defined on the matrix representations of the semigroup are interpreted in terms ofCPT conjugations. The theory of particles of spin one-half and systems with higher spin is developed from first principles. Methods are also developed for defining energy, momentum, orbital angular momentum, and weighted spacetime coordinates without reference to a manifold.

Some modules with nearly prescribed endomorphism rings

It is now well known that many, even quite small, rings can have large indecomposable modules or modules which have pathological decomposition properties. (For references, see [3].) The technique for obtaining such results is to construct modules with endomorphism rings of prescribed type. In this paper the technique is applied to rings with lattices of ideals satisfying a simple condition. However, the results obtained do not depend on the lattice condition since similar results hold for the integers (and other rings) which do not satisfy the condition. The present paper generalises a result of Dickson and Kelly [9]. These authors show that a ring R in which the identity has a decomposition 1 = e, + e2 + ... + e, into orthogonal idempotents such that eiRei is a local ring (1 < i < n) and which has nondistributive lattice of ideals has, for each cardinal c < N, , indecomposable modules with minimal generating set of cardinal c. The work described here arose from a resemblance between certain structures occurring in the work of Dickson and Kelly and some which occur in the study of endomorphism algebras of a vector space with four or more distinguished subspaces [I]. The author is very grateful to Professors Dickson and Kelly for communicating their results prior to publication. In addition she thanks Professor E. R. Love and the University of Melbourne for their hospitality in the Mathematics Department of the University of Melbourne at the time this work was started.

Some nondistributive lattices as initial segments of the degrees of unsolvability

The question, “What do initial segments of the degrees of unsolvability look like?” has interested recursive function theorists for several years. Sacks [4] hypothesized that Sis a finite initial segment of degrees if and only if S is order-isomorphic to a finite initial segment of some upper semilattice with a least element. Lachlan [2] suggested the generalization, S is an initial segment of degrees if S is order-isomorphic to some countable upper semilattice with both least and greatest elements.

An axiom system for the modular logic

G. Birkhoff and J. v. Neumann in their paper [4] concerning logic of the quantum mechanics come to the conclusion that not all laws of the Boolean algebra of logic apply to the statements of the quantum mechanics. Linear subspaces play an important part in the quantum mechanics. This fact suggested the authors the thought to con? sider the lattice of all linear subspaces of some linear space as a model of the sentential calculus of the quantum mechanics, analogously to the lattice of all subsets of a given set which can be considered as a model of the Boolean logic. Birkhof fand v. Neumann de? fine the modular logic as a free, modular and orthocomplementary lattice 3J? = freely generated by a set P. Elements belonging to P are called sentential variables, elements of M are called sentences, "v" and "a" are binary operations called alternative and conjunction respectively. " ' " is an unary operation called negation. One of the simplest models of the modular logic is the lattice of all linear subspaces of an ^-dimensional projective space. These subspaces are considered as sentences, the meet and linear union of two linear subspaces are considered as the conjunction and alternative of sentences. The orthogonal complementation of given subspace to the ^-dimensional space is referred to as negation. By a meet of two linear subspaces we understand their common part and by their linear union ? the smallest subspace containing both of them. In the modular logic the distributive law is abandoned. It is replaced by a weaker law called modular identity: (a a b) v (a a c) = a a (b v (a a c)). The modular logic must necessarily be an orthocomplementary lattice as in modular non-distributive lattices the complementarity relation is not a one-to-one relation, whereas the ortho complementarity is a relation of this kind [5]. It follows from the definition of the modular logic that there exist two fixed pro? positions denoted by 0 and 1. 0 is the false proposition, 1 is the true proposition. No implication operation is defined in the Birkhoff? v. Neumann modular lo? gic, however, the authors define an implication relation of partial order type. In the Boolean algebra "a implies b" if and only if "a v b = b". The implication is in the Boolean algebra uniquely defined as the operation a! ^ b (see [5]). This operation has in the modular logic few properties of the ordinary implication, e.g. a! v b = 1 does not imply the implication relation a v b = b.