PSEUDO-COMPLEMENTATION ON GENERALIZED ALMOST DISTRIBUTIVE LATTICES

2010 ◽  
Vol 03 (02) ◽  
pp. 335-346
Author(s):  
G. C. Rao ◽  
Ravi Kumar Bandaru ◽  
N. Rafi
Keyword(s):  
Distributive Lattice ◽  
Left Identity ◽  
Distributive Lattices ◽  
One To One ◽  
Identity Elements

The concept of a pseudo-complementation on a Generalized Almost Distributive Lattice (GADL) with 0 is introduced and proved that there is a one-to-one correspondence between the pseudo-complementations on a GADL L with 0 and left identity elements of L. The concept of *–GADL, disjunctive GADL is also introduced. We characterized *–GADL in terms of disjunctive GADL.

scholarly journals μ -Fuzzy Filters in Distributive Lattices

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Wondwosen Zemene Norahun
Keyword(s):  
Distributive Lattice ◽  
Special Class ◽  
Distributive Lattices ◽  
Fuzzy Filter ◽  
One To One ◽  
Fuzzy Filters

In this paper, we introduce the concept of μ -fuzzy filters in distributive lattices. We study the special class of fuzzy filters called μ -fuzzy filters, which is isomorphic to the set of all fuzzy ideals of the lattice of coannihilators. We observe that every μ -fuzzy filter is the intersection of all prime μ -fuzzy filters containing it. We also topologize the set of all prime μ -fuzzy filters of a distributive lattice. Properties of the space are also studied. We show that there is a one-to-one correspondence between the class of μ -fuzzy filters and the lattice of all open sets in X μ . It is proved that the space X μ is a T 0 space.

scholarly journals Almost distributive lattices

1981 ◽  
Vol 31 (1) ◽  
pp. 77-91 ◽  
Author(s):  
U. Maddana Swamy ◽  
G. C. Rao
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Boolean Ring ◽  
Triple Systems ◽  
Boolean Rings ◽  
Principal Ideals ◽  
Sheaf Representations ◽  
One To One ◽  
Almost All ◽  
Regular Rings

AbstractThe concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

scholarly journals Weak LI-ideals in implicative almost distributive lattices

2021 ◽  
Vol 14 (3) ◽  
pp. 207-217
Author(s):  
Tilahun Mekonnen Munie
Keyword(s):  
Distributive Lattice ◽  
Research Area ◽  
Distributive Lattices ◽  
Lattice Implication Algebra ◽  
Implication Algebra ◽  
New Development

In the field of many valued logic, lattice valued logic (especially ideals) plays an important role. Nowadays, lattice valued logic is becoming a research area. Researchers introduced weak LI-ideals of lattice implication algebra. Furthermore, other scholars researched LI-ideals of implicative almost distributive lattice. Therefore, the target of this paper was to investigate new development on the extension of LI-ideal theories and properties in implicative almost distributive lattice. So, in this paper, the notion of weak LI-ideals and maximal weak LI- ideals of implicative almost distributive lattice are defined. The properties of weak LI- ideals in implicative almost distributive lattice are studied and several characterizations of weak LI-ideals are given. Relationship between weak LI-ideals and weak filters are explored. Hence, the extension properties of weak LI-ideal of lattice implication algebra to that of weak LI-ideal of implicative almost distributive lattice were shown.

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 Finitely generated pseudocomplemented distributive lattices

1975 ◽  
Vol 19 (2) ◽  
pp. 238-246 ◽  
Author(s):  
J. Berman ◽  
ph. Dwinger
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Distributive Lattices ◽  
Basic Fact ◽  
Finitely Generated ◽  
Partially Ordered ◽  
Finite Set ◽  
Natural Way

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

Finite Projective Distributive Lattices

1970 ◽  
Vol 13 (1) ◽  
pp. 139-140 ◽  
Author(s):  
G. Grätzer ◽  
B. Wolk
Keyword(s):  
Distributive Lattice ◽  
Free Algebra ◽  
Distributive Lattices ◽  
Free Algebras ◽  
Algebra Ii

The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.

Metrizability Conditions for Completely Distributive Lattices

1983 ◽  
Vol 26 (4) ◽  
pp. 446-453
Author(s):  
G. Gierz ◽  
J. D. Lawson ◽  
A. R. Stralka
Keyword(s):  
Distributive Lattice ◽  
Essential Extension ◽  
Distributive Lattices ◽  
Interval Topology ◽  
Completely Distributive ◽  
Countable Chain ◽  
Completely Distributive Lattice

AbstractA lattice is said to be essentially metrizable if it is an essential extension of a countable lattice. The main result of this paper is that for a completely distributive lattice the following conditions are equivalent: (1) the interval topology on L is metrizable, (2) L is essentially metrizable, (3) L has a doubly ordergenerating sublattice, (4) L is an essential extension of a countable chain.

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 On the relation of a distributive lattice to its lattice of ideals

1972 ◽  
Vol 7 (3) ◽  
pp. 377-385 ◽  
Author(s):  
Herbert S. Gaskill
Keyword(s):  
Distributive Lattice ◽  
Related Result ◽  
Distributive Lattices ◽  
Relationship Of ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
Lattice Of Ideals

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

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