scholarly journals Uniqueness of representations of a distributive lattice as a free product of a Boolean algebra and a chain

1971 ◽  
Vol 24 (1) ◽  
pp. 27-35 ◽  
Author(s):  
R. Balbes ◽  
Ph. Dwinger
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Free Product ◽  
A Chain

scholarly journals Aleph–zero categorical Stone algebras

1978 ◽  
Vol 26 (3) ◽  
pp. 337-347 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Boolean Algebra ◽  
Characterization Problem ◽  
A Chain ◽  
Dense Set

AbstractThis paper is a contribution to the problem of characterizing the ℵ0-categorical Stone algebras. If the dense set is either finite or a chain, the problem is solved by reducing it to the ℵ0-categoricity of the skeleton and the dense set, solutions for these being known. If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras. For generalized Post algebras the characterization problem is solved completely.

scholarly journals Chains in generalized Boolean lattices

1976 ◽  
Vol 21 (2) ◽  
pp. 234-240
Author(s):  
Richard D. Byrd ◽  
Roberto A. Mena
Keyword(s):  
Distributive Lattice ◽  
Boolean Lattice ◽  
Boolean Lattices ◽  
A Chain

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

scholarly journals On Generalized Transitive Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Jing Jiang ◽  
Lan Shu ◽  
Xinan Tian
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Canonical Form ◽  
Transitive Closure ◽  
Fuzzy Algebra

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considered. Some properties of compositions of incline matrices are also given, and a new transitive incline matrix is constructed from given incline matrices. Finally, the issue of the canonical form of a transitive incline matrix is discussed. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

scholarly journals De Morgan Functions and Free De Morgan Algebras

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Yu. M. Movsisyan ◽  
V. A. Aslanyan ◽  
Alex Manoogian
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Functions ◽  
Bounded Lattice ◽  
De Morgan Algebra ◽  
Bounded Distributive Lattice ◽  
Monotone Boolean Functions ◽  
Free Generators ◽  
De Morgan Algebras

AbstractIt is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

scholarly journals On the order scale of a uniform space

1983 ◽  
Vol 34 (2) ◽  
pp. 248-257 ◽  
Author(s):  
D. C. Kent
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Closed Subspace ◽  
Uniform Space ◽  
Order Topology ◽  
Totally Bounded ◽  
Samuel Compactification ◽  
Closed Image ◽  
Atomic Boolean Algebra

AbstractThe order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

On the Triple Characterization for Stone Algebras

1975 ◽  
Vol 27 (4) ◽  
pp. 852-859 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Stone Algebra ◽  
Injective Hulls

In [1], C. C. Chen and G. Grâtzer developed a method for studying Stone algebras by associating with each Stone algebra L, a uniquely determined triple (C(L), D(L), ɸ (L)), consisting of a Boolean algebra C(L), a distributive lattice D(L), and a connecting map ɸ(L). This approach has been successfully exploited by various investigators to determine properties of Stone algebras (e.g. H. Lakser [9] characterized the injective hulls of Stone algebras by means of this technique). The present paper is a continuation of this program.

The Boolean Algebra of Regular Open Sets

1953 ◽  
Vol 5 ◽  
pp. 95-100 ◽  
Author(s):  
R. S. Pierce
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Distributive Lattice ◽  
Homomorphic Image ◽  
Continuous Functions ◽  
Completely Regular ◽  
Regular Open ◽  
Quasi Ordering

Let S be a completely regular topological space. Let C(S) denote the set of bounded, real-valued, continuous functions on 5. It is well known that C(S) forms a distributive lattice under the ordinary pointwise joins and meets. For any distributive lattice L and any ideal I⊆L, a quasi-ordering of L can be defined as follows : f⊇g if, for all h ∈ L, f ∩ h ∈ I implies g ∩ h ∈ I. If equivalent elements under this quasi-ordering are identified, a homomorphic image of L is obtained.

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