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# 数学代写|现代代数代考Modern Algebra代写|PARTIALLY ORDERED SETS

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## 数学代写|现代代数代考Modern Algebra代写|PARTIALLY ORDERED SETS

The first of several fundamental ideas for this chapter is an abstraction from $\leq$ for numbers and $\subseteq$ for sets.

Definition. A partially ordered set is a set $S$ together with a relation $\leq$ on $S$ such that each of the following axioms is satisfied:
Reflexive
If $a \in S$, then $a \leq a$.
Antisymmetric
If $a, b \in S, a \leq b$, and $b \leq a$, then $a=b$.
Transitive
If $a, b, c \in S, a \leq b$, and $b \leq c$, then $a \leq c$.
Example 63.1. The integers form a partially ordered set with respect to $\leq$. Here $\leq$ has its usual meaning. In other examples $\leq$ is replaced by whatever is appropriate for the relation involved.

Example 63.2. For each set $S$, let $P(S)$ denote the set of all subsets of $S$. Then $P(S)$ is a partially ordered set with $A \leq B$ defined to mean $A \subseteq B$. Notice that $\subseteq$ is a relation on $P(S)$, not on $S$. [The set $P(S)$ is called the power set of $S$; if $S$ is finite, then $|\mathcal{P}(S)|=2^{|S|}$, ” 2 to the power $|S| . “]$

Example 63.3. The set of all subgroups of any group is a partially ordered set with respect to $\subseteq$.

Example 63.4. The set $\mathbb{N}$ of all natural numbers (positive integers) is a partially ordered set with $a \leq b$ defined to mean $a \mid b$. The set of all positive divisors of a fixed positive integer $n$ is also a partially ordered set with this relation.

## 数学代写|现代代数代考Modern Algebra代写|LATTICES

Definition. A lattice is a partially ordered set in which each pair of elements has a least upper bound and a greatest lower bound.

The 1.u.b. of elements $a$ and $b$ in a lattice will be denoted $a \vee b$, and the g.l.b. will be denoted $a \wedge b$. The operations $\vee$ and $\wedge$ are called join and meet, respectively.

Example 64.1. The partially ordered sets in Examples 63.1 through 63.4 are all lattices. The proof for Example 63.1 is obvious, and proofs for the other cases follow from remarks in Section 63. We can now speak, for instance, of “the lattice of subgroups” of a group.

The definition of lattice demands that each pair of elements has a l.u.b. and a g.l.b. It follows from this that each finite subset has an l.u.b. and a g.l.b. For example, the l.u.b. of ${a, b, c}$ is $(a \vee b) \vee c$, which can be seen as follows. Let $u=(a \vee b) \vee c$. Then $a \vee b \leq u$, and therefore $a \leq u$ and $b \leq u$; also $c \leq u$. On the other hand, if $a \leq v, b \leq v$, and $c \leq v$, then $a \vee b \leq v$ and $c \leq v$, therefore $u=(a \vee b) \vee c \leq v$. Thus $u$ is a l.u.b. for ${a, b, c}$, as claimed. A similar argument shows that $(a \wedge b) \wedge c$ is a g.l.b. for ${a, b, c}$ (replace l.u.b. by g.l.b. and $\leq$ by $\geq$ throughout). The l.u.b. and g.l.b. of a finite subset $\left{a_1, a_2, \ldots, a_n\right}$ will be denoted
$$a_1 \vee a_2 \vee \cdots \vee a_n \quad \text { and } \quad a_1 \wedge a_2 \wedge \cdots \wedge a_n$$
respectively. The inductive proofs of the existence of these elements are left to Problem 64.16 .

In the paragraph preceding Example 63.7 we proved that if a subset of a partially ordered set has a l.u.b., then that l.u.b. is unique; we then stated that the uniqueness of the g.l.b. could be proved similarly: replace l.u.b. by g.l.b. and $\leq$ by $\geq$. In the same way, the proof that $(a \vee b) \vee c$ is an 1.u.b. for ${a, b, c}$ in a lattice, given previously, can be transformed into a proof that $(a \wedge b) \wedge c$ is a g.l.b. for ${a, b, c}$. These are applications of a very useful principle which, for lattices, has the following form.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|LATTICES

1.u.b。晶格中的元素$a$和$b$将表示为$a \vee b$，而g.l.b将表示为$a \wedge b$。操作$\vee$和$\wedge$分别称为join和meet。

$$a_1 \vee a_2 \vee \cdots \vee a_n \quad \text { and } \quad a_1 \wedge a_2 \wedge \cdots \wedge a_n$$

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## MATLAB代写

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