Projective Ordinal Sums of Lattices and Isotone Sections (original) (raw)
Related papers
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.
Isotone extensions and complete lattices
Ukraïnsʹkij matematičnij vìsnik, 2019
The set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an isotone extension to an isotone mapping from X to Y is found.
Two problems about perfect distributive lattices
Archiv der Mathematik, 1987
The study of the class P of perfect distributive lattices has been initiated in [2]. In this note we solve two problems left open in [2]. In the first section we give two subdirect representation theorems for the class Pol of perfect bounded distributive lattices (Theorems 1.4 and 1.8) and in the second one we determine those posets which are representable over P (Theorem 2.5). 0. Preliminaries. We assume familiarity with the elements of sheaf theory and the Priestley duality theory. For these topics we refer to [1] and [8], [5] and [6] respectively. Nevertheless, in order to make this paper more or less self-contained, we recall the main definitions and properties needed. All lattices that we consider in this paper are distributive and therefore the adjective "distributive" will generally be omitted. The notation D stands for the class of alldistributive-lattices. The subscript 0 (resp. 1) means that the lattices under consideration are bounded below (resp. above). A lattice Lisperfect if its prime spectrum, Spec L, (i. e., the set of its prime ideals ordered by inclusion) is the cardinal sum of chains. The class of all perfect lattices is denoted by P and the symbols P0, P1, P01 have an obvious meaning. The basic facts about perfect lattices are contained in [2]. A Boolean product (resp. weak Boolean product) of a family (A x [ x ~ X) of algebras over a Boolean space X is a subdirect product A of the given family such that the following conditions hold: (i) if a, b ~ A, then ~a = b~ = {x]ax = b~} is clopen (resp. open) in X; (it) if a, beA and Wis clopen in X, then a]w w b]_w~A. Obviously the definition of a weak Boolean product (resp. Boolean product) corresponds to sheaves (resp. Hausdorff sheaves) of algebras over a Boolean space [8]. A weak Boolean representation of a lattice L is an isomorphism O from L onto a weak Boolean product of lattices. As noticed in [3], we may always assume that the weak Boolean representations of non-trivial lattices are proper, that is, no stalk is trivial.
Projective and Injective Weak Distributive Lattice
2015
This Paper is Concerned with solving problems existence with quasi projective and quasi injective objects and retracts respectively over problems with projective and injective objects and retracts in the category whose objects are the complete quasi lattice and morphism are the complete quasi lattice homomorphism from the point of view .in this paper we mentioned here some necessary and sufficient conditions for the given lattice be quasi projective and quasi injective and retracts respect.
Lecture Notes in Mathematics, 1992
In this chapter we discuss some of the more recent results and give a general overview of what is currently known about lattice varieties. Of course it is impossible to give a comprehensive account. Often we only cite recent or survey papers, which themselves have many more references. We would like to apologize in advance for any errors, omissions, or miscrediting of results. For proofs of the results mentioned here, we refer the reader to the original papers. Details of many of the results from before 1992 can also be found in our monograph, P. Jipsen and H. Rose [39].