Subfitness in distributive (semi)lattices (original) (raw)

A Distributive Lattice Cover for Semilattices

2009

We consider two constructions of an envelope for a finite locally distributive strong upper semilattice. The first is based on Birkhoff's representation of finite distributive lattices and the second on valuations on lattices. We show that these produce isomorphic envelopes.

Maximal sublattices of finite distributive lattices

Algebra Universalis, 1996

Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0; 1)-lattices together with all (0; 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of nite distributive (0; 1)-lattices are thereby obtained.

Chapter VI: Finite Distributive Lattices

1983

We continue our study of the finite ideals of 2 in this chapter by showing that every finite distributive lattice is isomorphic to an ideal of Q>. This result is proved using techniques extending those introduced in Chap. V. Different trees are used, and we introduce tables which provide reduction procedures from the top degree of the ideal; these tables are obtained from representations of distributive lattices. As an application, we show that the set of minimal degrees forms an automorphism base for 2. Many of the applications which we obtain in later chapters from the complete characterization of the countable ideals of 2 can be obtained from the fact that all countable distributive lattices are isomorphic to ideals of 2. We use Exercise 4.17 of this chapter to indicate how to obtain the characterization of distributive ideals of 2. This exercise allows the reader to proceed directly to Chap. VIII.2 from the end of this chapter. The results of Appendix B.I are needed for this chapter.

On distributive join-semilattices

arXiv (Cornell University), 2019

Motivated by Gentzen's disjunction elimination rule in his Natural Deduction calculus and reading inequalities with meet in a natural way, we conceive a notion of distributivity for join-semilattices. We prove that it is equivalent to a notion present in the literature. In the way, we prove that those notions are linearly ordered. We finally consider the notion of distributivity in join-semilattices with arrow, that is, the algebraic structure corresponding to the disjunction-conditional fragment of intuitionistic logic. the use of that notion. Note that Hickman used the term mild distributivity for Hdistributivity. The paper is structured as follows. After this introduction, in Section 2 we provide some notions and notations that will be used in the paper. In Section 3 we show how to arrive to our notion of ND-distributivity for join-semilattices. In Section 4 we compare the different notions of distributivity for join-semilattices that appear in the literature. We prove that one of those is equivalent to the notion of ND-distributivity found in Section 3. Finally, in Section 5 we consider what happens with the different notions of distributivity considered in Section 4 when join-semilattices are expanded with a natural version of the relative meet-complement. In this section we provide the basic notions and notations that will be used in the paper. Let J = (J; ≤) be a poset. For any S ⊆ J, we will use the notations S l and S u to denote the set of lower and upper bounds of S, respectively. That is, S l = {x ∈ J : x ≤ s, for all s ∈ S} and S u = {x ∈ J : s ≤ x, for all s ∈ S}. Lemma 1. Let J = (J; ≤) be a poset. For all a, b, c ∈ J the following statements are equivalent: (i) for all x ∈ J, if x ≤ a and x ≤ b, then x ≤ c, (ii) {a, b} l ⊆ {c} l , (iii) c ∈ {a, b} lu . A poset J = (J; ≤) is a join-semilattice (resp. meet-semilattice) if sup{a, b} (resp. inf{a, b}) exists for every a, b ∈ J. A poset J = (J; ≤) is a lattice if it is both a join-and a meet-semilattice. As usual, the notations a ∨ b (resp. a ∧ b) shall stand for sup{a, b} (resp. inf{a, b}). Given a join-semilattice J = (J; ≤), we will use the following notions: • J is downwards directed iff for any a, b ∈ J, there exists c ∈ J such that c ≤ a and c ≤ b. • A non empty subset I ⊆ J is said to be an ideal iff (1) if x, y ∈ I, then x ∨ y ∈ I and (2) If x ∈ I and y ≤ x, then y ∈ I. • The principal ideal generated by an element a ∈ A, noted (a], is defined by (a] = {x ∈ A : x ≤ a}. • Id(J) will denote the set of all ideals of J. • Id f p (J) will denote the subset of ideals that are intersection of a finite set of principal ideals, that is, Id f p (J) = {(a 1 ] ∩ • • • ∩ (a k ] : a 1 , ...a k ∈ J}.

Some Characterizations of 0-DISTRIBUTIVE Semilattices

2012

In this paper we discuss prime down-sets of a semilattice. We give a characterization of prime down-sets of a semilattice. We also give some characterizations of 0-distributive semilattices and a characterization of minimal prime ideals containing an ideal of a 0-distributive semilattice. Finally, we give a characterization of minimal prime ideals of a pseudocomplemented semilattice.

Some Characterization of Directed Below 0-distributive Join Semilattices

Annals of Pure and Applied Mathematics

Varlet characterized the concept of 0-distributive lattices to generalize the notion of pseudo complemented lattices. Powar and Thakare introduced the notion of 0- Distributive semilattices. Very few authors have studied join semilattices. Among them Nimbhorkar and Rahemani, Akhter and Noor and Rao and Kumar have introduced the join semilattices. In this paper, we have introduced the notion of join semilattices with 0 - distributive directed below. A join semilattice S is called directed below if for any a,b ∈ S there is d ≤ a , b such that d ∈ S . Then obviously, every join semilattice with 0 is directed below

Some nonstandard methods applied to distributive lattices

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1990

In memory of Abraham Robinson, on the occasion of his 70th birthday Distributive lattices are locally finite algebras, i.e. their finitely generated sublattices are finite. The general theory of finite distributive lattices is remarkably transparent, due to one or both of the following principles: (i) Any filter (or ideal) is principal. (ii) The lattice is join-generated by its join-irreducible elements, namely by those elements z for which x v y = z implies x = z or y = z .