LATTICCESS AND BOOLEAN ALGEBRAS E-content for M. A/M.sc and Integrated Bsc-Msc Prepared by Dr Aftab Hussain Shah Assistant Professor Department of Mathematics JANUARY 1, 2105 CENTRAL UNIVERSITY OF KASHMIR Srinagar Contents Chapter-1: The Concept of an Order Chapter-2: Lattices Chapter-3: Modular Distributive and Complemented Lattices Chapter-4: Boolean Algebras and their applications Chapter 1 THE CONCEPT OF AN ORDER The present chapter is aimed at to provide introductory definitions, examples and results on ordered sets which shall be used in the subsequent chapters. This chapter consists of five sections. Section 1.1 is on ordered sets and their examples. In section 1.2 we discuss Hasse diagrams, a special type of diagrams used to represent ordered sets. Section 1.3 and 1.4 deals with results and examples on order preserving and residuated mappings between the ordered sets. The chapter ends with some important results and examples on isomorphism of ordered sets. 1.1 Concept of an order In this section we will go through the definition of partial order relation on a set, shortly read as “order” and discuss partial ordered sets (ordered sets) in detail with the help of various examples. The section ends with some of the important results on ordered sets. Definition 1.1.1: A binary relation on a non-empty set 퐸 is a subset 푅 of the Cartesian product set퐸 × 퐸 = {(푥, 푦) | 푥, 푦 ∈ 퐸}. For any (푥, 푦) ∈ 퐸 × 퐸, (푥, 푦) ∈ 푅 means that 푥 is related to 푦 under 푅 and we denote it by 푥푅푦. Definition 1.1.2: The Equivalence Relations on 퐸 is a binary relation 푅 that is: (a) Reflexive [for all푥 ∈ 퐸 (푥, 푥) ∈ 푅]; (b) Symmetric [for all 푥, 푦 ∈ 퐸 if (푥, 푦) ∈ 푅 then (푦, 푥) ∈ 푅]; (c) Transitive [for all 푥, 푦, 푧 ∈ 퐸if (푥, 푦) ∈ 푅 and (푦, 푧) ∈ 푅then (푥, 푧) ∈ 푅]. For any relation 푅 on 퐸 the dual of 푅 denoted by 푅푑 is defined by: (푥, 푦) ∈ 푅푑if and only if (푦, 푥) ∈ 푅. One can easily see that if 푅 is symmetric then 푅 = 푅푑. Here we shall be particularly interested in the situation where property (b) is replaced by the following property: (d) Antisymmetry[for all 푥, 푦 ∈ 퐸 if (푥, 푦) ∈ 푅 and (푦, 푥) ∈ 푅 then 푥 = 푦]. 푑 One can easily verify that if 푅is antisymmetric then 푅 ∩ 푅 = 푖푑퐸. Where 푖푑퐸 denotes the equality relation on 퐸as: (푥, 푦) ∈ 푅 ∩ 푅푑 if and only if (푥, 푦) ∈ 푅 and (푥, 푦) ∈ 푅푑; if and only if (푥, 푦) ∈ 푅 and (푦, 푥) ∈ 푅; if and only if 푥 = 푦; (since 푅 is antisymmetric) if and only if (푥, 푦) ∈ 푖푑퐸; 푑 if and only if 푅 ∩ 푅 = 푖푑퐸. Notation: We usually denote 푅 by the symbol ≤ and write the expression (푥, 푦) ∈ ≤ in the equivalent form 푥 ≤ 푦. Which we read as “푥 is less than or equal to 푦”. With this notation we now define an order on a set 퐸. Definition 1.1.3: We say ≤ is an order on 퐸 if and only if: (a) Reflexivity: [For all푥 ∈ 퐸 푥 ≤ 푥.] (b) Antisymmetry: [For all푥, 푦 ∈ 퐸 if 푥 ≤ 푦 and 푦 ≤ 푥 then 푥 = 푦.] (c) Transitivity:[For all 푥, 푦, 푧 ∈ 퐸 if 푥 ≤ 푦 and 푦 ≤ 푧 then 푥 ≤ 푧.] Definition 1.1.4: By an ordered set we shall mean a set 퐸 on which there is defined an order ≤ and we denote it by (퐸; ≤). Other common terminology for an order is a partial order and for an ordered set is a partially ordered set or a poset. Example 1.1.5: On every set 퐸 the relation of equality is an order. Solution: Let 퐸 be any arbitrary set. Suppose for all 푥, 푦 ∈ 퐸 푥 ≤ 푦 if and only if 푥 = 푦. Reflexivity: For all 푥 ∈ 퐸 we have 푥 = 푥. So푥 ≤ 푥 for all 푥 ∈ 퐸. Antisymmetry: For any 푥, 푦 ∈ 퐸 suppose 푥 ≤ 푦 and 푦 ≤ 푥 if and only if 푥 = 푦 (by definitionof ≤). Thus ≤ is antisymmetric. Transitivity: For any 푥, 푦, 푧 ∈ 퐸 let 푥 ≤ 푦 and 푦 ≤ 푧 if and only if 푥 = 푦 and 푦 = 푧 if and only if 푥 = 푧 if and only if 푥 ≤ 푧. Which proves that ≤ is transitive. Example 1.1.6: On the set ℙ(퐸) of all subsets of a non-empty set 퐸, the relation ⊆ of set inclusion defined by 퐴 ≤ 퐵 if and only if 퐴 ⊆ B is an order. Solution: Reflexivity: For any 퐴 ∈ ℙ(퐸), since 퐴 ⊆ A so 퐴 ≤ 퐴. Therefore ‘⊆’ is reflexive. Antisymmetry: For any 퐴, 퐵 ∈ 푃(퐸) let퐴 ≤ 퐵 and 퐵 ≤ 퐴 implies 퐴 ⊆ B and 퐵 ⊆ A; this gives 퐴 = 퐵. Therefore ⊆ is antisymmetric. Transitivity: For any 퐴, 퐵, 퐶 ∈ 푃(퐸) let 퐴 ≤ 퐵 and 퐵 ≤ 퐶, which implies 퐴 ⊆ B and 퐵 ⊆ C; this gives 퐴 ⊆ 퐶; therefore 퐴 ≤ 퐶 and thus ⊆ is transitive. Hence ⊆ defines an order on ℙ(퐸). Example 1.1.7: On the set ℕ of natural numbers the relation | of divisibility, defined by 푚 ≤ 푛 if and only if 푚|푛, is an order. Solution: To show that ℕ with respect to | forms an order set we must show that it satisfies reflexivity, antisymmetry and transitivity. Reflexivity: For any 푚 ∈ ℕ, we have 푚 = 1 ∙ 푚, so 푚|푚, thus 푚 ≤ 푚 and hence ≤ is reflexive Antisymmetry: For any 푚, 푛 ∈ ℕ suppose 푚 ≤ 푛 and 푛 ≤ 푚 if and only if 푚|푛 and 푛|푚 if and only if 푚 = 푛. Thus ≤ is antisymmetric. Transitivity: For 푚, 푛, 푝 ∈ ℕ suppose 푚 ≤ 푛 and 푛 ≤ 푝 if and only if 푚|푛 and 푛|푝 if and only if 푚|푝 i.e, 푚 ≤ 푛. Thus ≤ is transitive. So| is an order on ℕ. Example 1.1.8: If (푃; ≤ ) is an ordered set and 푄 is a subset of 푃 then the relation≤푄 defined on 푄 by; 푥 ≤푄 푦 if and only if 푥 ≤ 푦 is an order on 푄. Solution: We need to show that the relation ≤푄 defined on 푄 is an order. Reflexivity: For any 푥 ∈ 푄, we have 푥 ∈ 푃(푄 ⊆ 푃), since푃 is an ordered set, therefore we have 푥 ≤ 푥 if and only if 푥 ≤푄 푥. Thus ≤푄 is reflexive on 푄. Antisymmetry: Let 푥, 푦 ∈ 푄 be such that 푥 ≤푄 푦 and 푦 ≤푄 푥; if and only if 푥 ≤ 푦and 푦 ≤ 푥 if and only if 푥 = 푦 (since ≤ is an order on 푃). This proves that ≤푄 is antisymmetric. Transitivity: Suppose for any 푥, 푦, 푧 ∈ 푄; 푥 ≤푄 푦 and 푦 ≤푄 푧; if and only if 푥 ≤ 푦 and 푦 ≤ 푧 if and only if 푥 ≤ 푧 (since ≤ is an order on 푃) if and only if 푥 ≤푄 푧. This proves that ≤푄 is transitive. We often write ≤푄 simply as ≤ and say 푄 inherits the order ≤ from 푃. Thus for example, the set 퐸푞푢 퐸 of all equivalence relations on 퐸 inherits the order ⊆ from 푃(퐸 × 퐸). Example 1.1.9: The set of even positive integers may be ordered in a usual way or by divisibility. Example 1.1.10: If (퐸1; ≤1), (퐸2; ≤2), . , (퐸푛; ≤푛) are ordered sets then the Cartesian 푛 product set×푖=1 퐸i can be given the Cartesian order ≤ defined by: for any (푥1, 푥2, . , 푥푛), 푛 (푦1, 푦2, . , 푦푛) ∈ ×푖=1 퐸푖; (푥1, 푥2, . , 푥푛) ≤ (푦1, 푦2, . , 푦푛) if and only if 푥푖 ≤푖 푦푖 for all 푖 = 1,2, … , 푛. 푛 Solution: Reflexivity: Let (푥1, 푥2, . , 푥푛) ∈×푖=1 퐸푖 where 푥푖 ∈ 퐸푖 for all 푖 = 1,2, … , 푛. Since (퐸1; ≤1) , (퐸2; ≤2), . , (퐸푛; ≤푛) are ordered sets then by reflexivity of each (퐸푖; ≤푖) 푥푖 ≤푖 푥푖 for all 푖 = 1,2, … , 푛 if and only if (푥1, 푥2, . , 푥푛) ≤ (푥1, 푥2, . , 푥푛) (by definition of ≤). This Shows that ≤ is reflexive. 푛 Antisymmetry: Let (푥1, 푥2, . , 푥푛), (푦1, 푦2, . , 푦푛) ∈ ×푖=1 퐸푖 where 푥푖 ∈ 퐸푖 and 푦푖 ∈ 퐸푖 for all 푖 = 1,2, … , 푛. Suppose (푥1, 푥2, . , 푥푛) ≤ (푦1, 푦2, . , 푦푛) and (푦1, 푦2, . , 푦푛) ≤ (푥1, 푥2, . , 푥푛) if and only if 푥푖 ≤푖 푦푖 and 푦푖 ≤푖 푥푖 for all 푖 = 1,2, … , 푛 if and only if 푥푖 = 푦푖 for all 푖 = 1,2, … , 푛 (by antisymmetry of each (퐸푖; ≤푖)) if and only if (푥1, 푥2, . , 푥푛) = (푦1, 푦2, . , 푦푛). Which showsthat ≤ is antisymmetric. 푛 Transitivity: For any (푥1, 푥2, . , 푥푛), (푦1, 푦2, . , 푦푛) and (푧1, 푧2, . , 푧푛) ∈×푖=1 퐸푖 if (푥1, 푥2, . , 푥푛) ≤ (푦1, 푦2, . , 푦푛) and (푦1, 푦2, . , 푦푛) ≤ (푧1, 푧2, . , 푧푛) if and only if 푥푖 ≤푖 푦푖 and 푦푖 ≤푖 푧푖 for all 푖 = 1,2, … , 푛 if and only if 푥푖 ≤푖 푧푖 for all 푖 = 1,2, … , 푛 (by transitivity of each (퐸푖; ≤푖)) if and only if (푥1, 푥2, . , 푥푛) ≤ (푧1, 푧2, . , 푧푛), proving that ≤ is transitive. Definition 1.1.11: The order defined in above is called the Cartesian order. Example 1.1.12: Let 퐸 and 퐹 be ordered sets, then the set 푀푎푝(퐸, 퐹) of all mappings 푓: 퐸 → 퐹 can be ordered by defining 푓 ≤ 푔 if and only if 푓(푥) ≤ 푔(푥) for all 푥 ∈ 퐸. Solution: Reflexivity: Let 푥 ∈ 퐸, then푓(푥) ∈ 퐹, since 퐹 is an ordered set we have 푓(푥) ≤ 푓(푥) if and only if 푓 ≤ 푓 showing that (푀푎푝 (퐸, 퐹), ≤ ) is reflexive. Antisymmetry: For any 푓, 푔 ∈ (푀푎푝 (퐸, 퐹), ≤ ) suppose 푓 ≤ 푔 and 푔 ≤ 푓; if and only if 푓(푥) ≤ 푔(푥) and 푔(푥) ≤ 푓(푥) for all 푥 ∈ 퐸, since 푓(푥), 푔(푥) ∈ 퐹 and 퐹 is an ordered set therefore by antisymmetry of 퐹 we have 푓(푥) = 푔(푥) for any 푥 ∈ 퐸, thus 푓 ≤ 푔 and푔 ≤ 푓 implies 푓 = 푔.