scholarly journals A Bijection for Essentially 4-Connected Toroidal Triangulations

10.37236/7897 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Nicolas Bonichon ◽  
Benjamin Lévêque
Keyword(s):  
Distributive Lattice ◽  
Minimal Element ◽  
Distributive Lattices ◽  
Outer Face ◽  
Random Generation ◽  
Combinatorial Structures ◽  
Toroidal Triangulations

Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications (drawing algorithm, random generation, enumeration, ...). In this paper we introduce and study a generalization of these objects for the toroidal case. Contrary to what happens in the plane, the set of toroidal transversal structures of a given toroidal triangulation is partitioned into several distributive lattices. We exhibit a subset of toroidal transversal structures, called balanced, and show that it forms a single distributive lattice. Then, using the minimal element of the lattice, we are able to enumerate bijectively essentially 4-connected toroidal triangulations.

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′).

Frontal operators in distributive lattices with a generalized implication

2015 ◽  
Vol 08 (03) ◽  
pp. 1550039
Author(s):  
Sergio A. Celani ◽  
Hernán J. San Martín
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Duke Math ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Bounded Distributive Lattice ◽  
Modal Lattices ◽  
Weak Heyting Algebras ◽  
San Martín

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

scholarly journals Minimum sized fibres in distributive lattices

2001 ◽  
Vol 70 (3) ◽  
pp. 337-350 ◽  
Author(s):  
Dwight Duffus ◽  
Bill Sands
Keyword(s):  
Lower Bound ◽  
Distributive Lattice ◽  
Ordered Set ◽  
The Other ◽  
Distributive Lattices ◽  
Maximal Antichain ◽  
Monotone Property ◽  
Fibre Size

AbstractA subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.

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