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.

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:

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.

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 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).

MS-almost distributive lattices

2019 ◽  
Vol 13 (07) ◽  
pp. 2050135
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Equational Class ◽  
Distributive Lattices

The purpose of this paper is to define and investigate a new equational class of algebras which we call MS-almost distributive lattices (MS-ADLs) as a common abstraction of De Morgan ADLs and Stone ADLs. It is observed that the class of MS-ADLs properly contains the class of MS-algebras, and most of the properties of MS-algebras are extended to the class of MS-ADLs.

scholarly journals Lattices whose ideal lattice is Stone

1983 ◽  
Vol 26 (1) ◽  
pp. 107-112 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Distributive Lattice ◽  
Dual Space ◽  
Sufficient Conditions ◽  
Distributive Lattices ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Topological Duality ◽  
Bounded Distributive Lattice ◽  
Image Position ◽  
The Ideal

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chainof type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.

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.

Postulates For Distributive Lattices

1951 ◽  
Vol 3 ◽  
pp. 28-30 ◽  
Author(s):  
Marlow Sholander
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Lattice Theory ◽  
Distributive Lattices ◽  
Intrinsic Interest ◽  
Binary Operations ◽  
The Third ◽  
Ternary Operation

Many sets of postulates have been given for distributive lattices and for Boolean algebra. For a description of some of the most interesting and for references to others the reader is referred to Birkhoff's “Lattice Theory”[1]. In this paper we give sets of postulates which have some intrinsic interest because of their simplicity. In the first two sections binary operations are used to describe a distributive lattice by 2 identities in 3 variables and a Boolean algebra by 3 identities in 3 variables. In the third section a ternary operation is used to describe distributive lattices with 0 and J by 2 identities in 5 variables.

scholarly journals Pseudocomplemented distributive lattices with small endomorphism monoids

1983 ◽  
Vol 28 (3) ◽  
pp. 305-318 ◽  
Author(s):  
M.E. Adams ◽  
V. Koubek ◽  
J. Sichler
Keyword(s):  
The Other ◽  
Distributive Lattices ◽  
Endomorphism Monoid ◽  
Proper Class ◽  
Lattice Of Varieties ◽  
Endomorphism Monoids ◽  
Other Hand ◽  
Image Position

By a result of K.B. Lee, the lattice of varieties of pseudo-complemented distributive lattices is the ω + 1 chainwhere B−1, B0, B1 are the varieties formed by all trivial, Boolean, and Stone algebras, respectively. General theorems on relative universality proved in the present paper imply that there is a proper class of non-isomorphic algebras in B3 with finite endomorphism monoids, while every infinite algebra from B2 has infinitely many endomorphisms. The variety B4 contains a proper class of non-isomorphic algebras with endomorphism monoids consisting of the identity and finitely many right zeros; on the other hand, any algebra in B3 with a finite endomorphism monoid of this type must be finite.

A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees

1969 ◽  
Vol 21 ◽  
pp. 147-148 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Image Position

In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem.THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees.For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j:12Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.

Sign in / Sign up

Export Citation Format

Share Document