Distributive Projective Lattices

1970 ◽  
Vol 22 (3) ◽  
pp. 472-475 ◽  
Author(s):  
Kirby A. Baker ◽  
Alfred W. Hales
Keyword(s):  
Lattice Theory ◽  
Free Lattice ◽  
Important Case ◽  
Distributive Lattices ◽  
Interesting Corollary ◽  
Projective Lattices

Two basic unsolved problems of lattice theory are (1) the characterization of sublattices of free lattices and (2) the characterization of projective lattices. A solution to an important case of the first problem has been provided by Galvin and Jónsson [3], who characterize distributive sublattices of free lattices. In this paper, we solve the same case of the second problem by characterizing distributive projective lattices (Theorem 4.1). An interesting corollary is the verification for distributive lattices of the conjecture that a, finite lattice is projective if and only if it is a sublattice of a free lattice.

scholarly journals Stone algebras form an equational class: (Remarks on Lattice Theory III)

1969 ◽  
Vol 9 (3-4) ◽  
pp. 308-309 ◽  
Author(s):  
G. Grätzer
Keyword(s):  
Lattice Theory ◽  
Equational Class ◽  
Distributive Lattices ◽  
Image Position

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

scholarly journals Congruences on a Distributive Lattice

1954 ◽  
Vol 10 (2) ◽  
pp. 76-77
Author(s):  
H. A. Thueston
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Distributive Lattices ◽  
The Subject ◽  
The Many ◽  
Finite Distributive Lattices

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

Distributive Sublattices of a Free Lattice

1961 ◽  
Vol 13 ◽  
pp. 265-272 ◽  
Author(s):  
Fred Galvin ◽  
Bjarni Jónsson
Keyword(s):  
Free Lattice ◽  
Distributive Lattices

The purpose of this note is to characterize those distributive lattices that can be isomorphically embedded in free lattices. If it is known (cf. (2)) that in a free lattice every element is either additively or multiplicatively irreducible, and consequently every sublattice of a free lattice must also have this property. We therefore begin by studying the class of all those distributive lattices in which this condition is satisfied.

On the Structure of Finitely Presented Lattices

1981 ◽  
Vol 33 (2) ◽  
pp. 404-411 ◽  
Author(s):  
G. Gratzer ◽  
A. P. Huhn ◽  
H. Lakser
Keyword(s):  
Congruence Relation ◽  
Lattice Theory ◽  
Structure Theorem ◽  
Congruence Class ◽  
Free Lattice ◽  
Partial Lattice ◽  
Finitely Presented

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.In this paper we prove a structure theorem that could be said to verify this conjecture.THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

scholarly journals BOND-FREE LANGUAGES: FORMALIZATIONS, MAXIMALITY AND CONSTRUCTION METHODS

2005 ◽  
Vol 16 (05) ◽  
pp. 1039-1070 ◽  
Author(s):  
LILA KARI ◽  
STAVROS KONSTANTINIDIS ◽  
PETR SOSÍK
Keyword(s):  
Polynomial Time ◽  
Structural Characterization ◽  
Regular Languages ◽  
Important Case ◽  
Similarity Relation ◽  
Construction Methods ◽  
Large Sets ◽  
Code Property ◽  
Context Free

The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.

Yap Hian Poh. Postulational study of an axiom system of Boolean algebra. Majallah Tahunan 'Ilmu Pasti—Shu Hsüeh Nien K'an—Bulletin of Mathematical Society of Nanyang University (1960), pp. 94–110. - R. M. Dicker. A set of independent axioms for Boolean algebra. Proceedings of the London Mathematical Society, ser. 3 vol. 13 (1963), pp. 20–30. - P. J. van Albada. A self-dual system of axioms for Boolean algebra. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, series A vol. 67 (1964), pp. 377–381; also Indagationes mathematicae, vol. 26 (1964), pp. 377–381. - Antonio Diego and Alberto Suárez. Two sets of axioms for Boolean algebras. Portugaliae mathematica, vol. 23 nos. 3–4 (for 1964, pub. 1965), pp. 139–145. (Reprinted from Notas de lógica matemática no. 16, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca 1964, 13 pp.) - P. J. van Albada. Axiomatique des algèbres de Boole. Bulletin de la Société Mathématique de Belgique, vol. 18 (1966), pp. 260–272. - Lawrence J. Dickson. A short axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 6 (1967), pp. 253–257. - Leroy J. Dickey. A shorter axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 8 (1968), p. 336. - Chinthayamma . Independent postulate sets for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 9 (1968), pp. 378–379. - Kiyoshi Iséki. A simple characterization of Boolean rings. Proceedings of the Japan Academy, vol. 44 (1968), pp. 923–924. - Sakiko Ôhashi. On definitions of Boolean rings and distributive lattices. Proceedings of the Japan Academy, vol. 44 (1968), pp. 1015–1017.

1973 ◽  
Vol 38 (4) ◽  
pp. 658-660
Author(s):  
Donald H. Potts
Keyword(s):  
Boolean Algebra ◽  
London Mathematical Society ◽  
Axiom System ◽  
Dual System ◽  
Boolean Algebras ◽  
Axiomatic System ◽  
Mathematical Society ◽  
Distributive Lattices ◽  
Boolean Rings

scholarly journals Formalization of Generalized Almost Distributive Lattices

2014 ◽  
Vol 22 (3) ◽  
pp. 257-267
Author(s):  
Adam Grabowski
Keyword(s):  
Boolean Algebra ◽  
Lattice Theory ◽  
Sufficient Conditions ◽  
Specific Area ◽  
Zero Element ◽  
Distributive Lattices ◽  
Starting Point ◽  
Good Starting Point ◽  
Mizar Mathematical Library ◽  
Necessary And Sufficient

Summary Almost Distributive Lattices (ADL) are structures defined by Swamy and Rao [14] as a common abstraction of some generalizations of the Boolean algebra. In our paper, we deal with a certain further generalization of ADLs, namely the Generalized Almost Distributive Lattices (GADL). Our main aim was to give the formal counterpart of this structure and we succeeded formalizing all items from the Section 3 of Rao et al.’s paper [13]. Essentially among GADLs we can find structures which are neither V-commutative nor Λ-commutative (resp., Λ-commutative); consequently not all forms of absorption identities hold. We characterized some necessary and sufficient conditions for commutativity and distributivity, we also defined the class of GADLs with zero element. We tried to use as much attributes and cluster registrations as possible, hence many identities are expressed in terms of adjectives; also some generalizations of wellknown notions from lattice theory [11] formalized within the Mizar Mathematical Library were proposed. Finally, some important examples from Rao’s paper were introduced. We construct the example of GADL which is not an ADL. Mechanization of proofs in this specific area could be a good starting point towards further generalization of lattice theory [10] with the help of automated theorem provers [8].

scholarly journals Algorithm and Parameters: Solving the Generality Problem for Reliabilism

2019 ◽  
Vol 128 (4) ◽  
pp. 463-509 ◽  
Author(s):  
Jack C. Lyons
Keyword(s):  
Case Studies ◽  
Cognitive Processes ◽  
Folk Psychology ◽  
Important Case ◽  
Cognitive Penetration ◽  
Generality Problem ◽  
Psychological Variables ◽  
Affect Processing

The paper offers a solution to the generality problem for a reliabilist epistemology, by developing an “algorithm and parameters” scheme for type-individuating cognitive processes. Algorithms are detailed procedures for mapping inputs to outputs. Parameters are psychological variables that systematically affect processing. The relevant process type for a given token is given by the complete algorithmic characterization of the token, along with the values of all the causally relevant parameters. The typing that results is far removed from the typings of folk psychology, and from much of the epistemology literature. But it is principled and empirically grounded, and shows good prospects for yielding the desired epistemological verdicts. The paper articulates and elaborates the theory, drawing out some of its consequences. Toward the end, the fleshed-out theory is applied to two important case studies: hallucination and cognitive penetration of perception.

scholarly journals On Two Alternative Axiomatizations of Lattices by McKenzie and Sholander

2018 ◽  
Vol 26 (2) ◽  
pp. 193-198
Author(s):  
Adam Grabowski ◽  
Damian Sawicki
Keyword(s):  
Lattice Theory ◽  
Source Code ◽  
Distributive Lattices ◽  
Formal Proofs ◽  
Mizar Mathematical Library

Summary The main result of the article is to prove formally that two sets of axioms, proposed by McKenzie and Sholander, axiomatize lattices and distributive lattices, respectively. In our Mizar article we used proof objects generated by Prover9. We continue the work started in [7], [21], and [13] of developing lattice theory as initialized in [22] as a formal counterpart of [11]. Complete formal proofs can be found in the Mizar source code of this article available in the Mizar Mathematical Library (MML).

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