Projective Ordinal Sums of Lattices and Isotone Sections (original) (raw)
Pacific Journal of Mathematics, 1978
This paper gives necessary and sufficient conditions for a lattice to be projective. The conditions are the Whitman condition, and a condition of Jόnsson, and two new conditions explained below.
R A ] 1 2 D ec 2 01 8 Ordinal and Horizontal Sums Constructing PBZ ∗ – Lattices
2018
PBZ∗–lattices are algebraic structures related to quantum logics, which consist of bounded lattices endowed with two kinds of complements, named Kleene and Brouwer, such that the Kleene complement satisfies a weakening of the orthomodularity condition and the De Morgan laws, while the Brouwer complement only needs to satisfy the De Morgan laws for the pairs of elements with their Kleene complements. PBZ∗–lattices form a variety PBZL∗, which includes the variety OML of orthomodular lattices (considered with an extended signature, by letting their two complements coincide) and the variety V (AOL) generated by the class AOL of antiortholattices. We investigate the congruences of antiortholattices, in particular of those obtained through certain ordinal sums and of those whose Brower complements satisfy the De Morgan laws, infer characterizations for their subdirect irreducibility and prove that even the lattice reducts of antiortholattices are directly irreducible. Since the two comple...
Ordinal and Horizontal Sums Constructing PBZ*-lattices
arXiv: Rings and Algebras, 2018
PBZ*-lattices are algebraic structures related to quantum logics, which consist of bounded lattices endowed with two kinds of complements, named {\em Kleene} and {\em Brouwer}, such that the Kleene complement satisfies a weakening of the orthomodularity condition and the De Morgan laws, while the Brouwer complement only needs to satisfy the De Morgan laws for the pairs of elements with their Kleene complements. PBZ*-lattices form a variety mathbbPBZLast\mathbb{PBZL}^{\ast }mathbbPBZLast, which includes the variety mathbbOML\mathbb{OML}mathbbOML of orthomodular lattices (considered with an extended signature, by letting their two complements coincide) and the variety V(mathbbAOL)V(\mathbb{AOL})V(mathbbAOL) generated by the class mathbbAOL\mathbb{AOL}mathbbAOL of antiortholattices. We investigate the congruences of antiortholattices, in particular of those obtained through certain ordinal sums, and infer characterizations for their subdirect irreducibility. We also prove that antiortholattices are the directly irreducible members of the variety they generate, a...
An Analogue of Distributivity for Ungraded Lattices
Order, 2006
In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Other than distributive lattices, the main examples of trim lattices are the Tamari lattices and various generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim, and we conjecture that all Cambrian lattices are trim.
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 conjectures in orthocomplemented lattices
Artificial Intelligence, 2000
A mathematical model for conjectures in orthocomplemented lattices is presented. After defining when a conjecture is a consequence or a hypothesis, some operators of conjectures, consequences and hypotheses are introduced and some properties they show are studied. This is the case, for example, of being monotonic or non-monotonic operators. As orthocomplemented lattices contain orthomodular lattices and Boolean algebras, they offer a sufficiently broad framework to obtain some general results that can be restricted to such particular, but important, lattices. This is, for example, the case of the structure's theorem for hypotheses. Some results are illustrated by examples of mathematical or linguistic character, and an appendix on orthocomplemented lattices is included.
A characterization of well-founced algebraic lattices
Contributions Discret. Math., 2018
We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join semi-lattice of compact elements of L, is well founded and contains neither [\omega]^\omega, nor \underscore(\Omega)(\omega*) as a join semilattice. As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is well founded and contains no infinite independent set. If K(L) is a join-subsemilattice of I_{<\omega}(Q), the set of finitely generated initial segments of a well founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered.
On values in relatively normal lattices
Discrete Mathematics, 1996
In [8, 11, 12] the class IRN was introduced in order to obtain the lattice-theoretic analogues of some results of Conrad (see e.g. [4]). The aim of these paper is to provide other useful constructions in the study of the structure of relatively normal lattices. The introduced notions and results are purely lattice-theoretic extensions of notions and results for lattice-ordered groups [2, 4, 5]. In the second section, the notion of plenary set of a member of the class IRN is introduced and the characterization of maximal plenary sets is given, extending a well-known theorem in/-groups. In the third section with any lattice in IRN is associated a tree and we investigate how the properties of this tree are reflected in the structure of the lattice. For the case of/-groups, one gets some of Conrad's results in [5].