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Lattices

We begin the study of lattices and Boolean algebras by generalizing the idea of inequality. Recall that a relation on a set X is a subset of X \times X.

Partially Ordered Sets

We begin the study of lattices and Boolean algebras by generalizing the idea of inequality. Recall that a relation on a set \(X\) is a subset of \(X \times X\). A relation \(P\) on \(X\) is called a partial order of \(X\) if it satisfies the following axioms.

  1. The relation is reflexive: \((a, a) \in P\) for all \(a \in X\).

  2. The relation is antisymmetric: if \((a,b) \in P\) and \((b,a) \in P\), then \(a = b\).

  3. The relation is transitive: if \((a, b) \in P\) and \((b, c) \in P\), then \((a, c) \in P\).

We will usually write \(a \preceq b\) to mean \((a, b) \in P\) unless some symbol is naturally associated with a particular partial order, such as \(a \leq b\) with integers \(a\) and \(b\), or \(A \subset B\) with sets \(A\) and \(B\). A set \(X\) together with a partial order \(\preceq\) is called a partially ordered set, or poset. \(a \preceq b\) \(a\) is less than \(b\)

Example

The set of integers (or rationals or reals) is a poset where \(a \leq b\) has the usual meaning for two integers \(a\) and \(b\) in \({\mathbb Z}\).

Example

Let \(X\) be any set. We will define the power set of \(X\) to be the set of all subsets of \(X\). We denote the power set of \(X\) by \({\mathcal P}(X)\). For example, let \(X = \{ a, b, c \}\). Then \({\mathcal P}(X)\) is the set of all subsets of the set \(\{ a, b, c \}\): \[\begin{aligned}& \emptyset & & \{ a \} & & \{ b \} & & \{ c \} & \\ & \{ a, b \} & & \{ a, c\} & &\{ b, c\} & & \{ a, b, c \}. &\end{aligned}\] On any power set of a set \(X\), set inclusion, \(\subset\), is a partial order. We can represent the order on \(\{ a, b, c \}\) schematically by a diagram such as the one in .

Example

Let \(G\) be a group. The set of subgroups of \(G\) is a poset, where the partial order is set inclusion.

Example

There can be more than one partial order on a particular set. We can form a partial order on \({\mathbb N}\) by \(a \preceq b\) if \(a \mid b\). The relation is certainly reflexive since \(a \mid a\) for all \(a \in {\mathbb N}\). If \(m \mid n\) and \(n \mid m\), then \(m = n\); hence, the relation is also antisymmetric. The relation is transitive, because if \(m \mid n\) and \(n \mid p\), then \(m \mid p\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

Pyetja që bëjnë njerëzit

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Pjesa e kësaj faqeje është adaptuar nga Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). E përmbledhur dhe ri-shkruar këtu; gabimet janë tona.

Më shumë në Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula