Abstract Algebra: Theory and Applications

Draw the lattice diagram for the power set of $X=\{a,b,c,d\}$ with the set inclusion relation, $\subset \text{.}$

Draw the diagram for the set of positive integers that are divisors of $30\text{.}$ Is this poset a Boolean algebra?

Draw a diagram of the lattice of subgroups of ${\mathbb{Z}}_{12}\text{.}$

Let $B$ be the set of positive integers that are divisors of $210\text{.}$ Define an order on $B$ by $a⪯b$ if $a\mid b\text{.}$ Prove that $B$ is a Boolean algebra. Find a set $X$ such that $B$ is isomorphic to $\mathcal{P}(X)\text{.}$

Prove or disprove: $\mathbb{Z}$ is a poset under the relation $a⪯b$ if $a\mid b\text{.}$

Draw a circuit that will be closed exactly when only one of three switches $a\text{,}$ $b\text{,}$ and $c$ are closed.

Prove or disprove that the two circuits shown are equivalent.

Let $X$ be a finite set containing $n$ elements. Prove that $|\mathcal{P}(X)|=2^n\text{.}$ Conclude that the order of any finite Boolean algebra must be $2^n$ for some $n\in \mathbb{N}\text{.}$

For each of the following circuits, write a Boolean expression. If the circuit can be replaced by one with fewer switches, give the Boolean expression and draw a diagram for the new circuit.

Prove or disprove: The set of all nonzero integers is a lattice, where $a⪯b$ is defined by $a\mid b\text{.}$

Let $L$ be a nonempty set with two binary operations $\vee$ and $\wedge$ satisfying the commutative, associative, idempotent, and absorption laws. We can define a partial order on $L\text{,}$ as in Theorem 19.14, by $a⪯b$ if $a\vee b=b\text{.}$ Prove that the greatest lower bound of $a$ and $b$ is $a\wedge b\text{.}$

Let $G$ be a group and $X$ be the set of subgroups of $G$ ordered by set-theoretic inclusion. If $H$ and $K$ are subgroups of $G\text{,}$ show that the least upper bound of $H$ and $K$ is the subgroup generated by $H\cup K\text{.}$

Let $R$ be a ring and suppose that $X$ is the set of ideals of $R\text{.}$ Show that $X$ is a poset ordered by set-theoretic inclusion, $\subset \text{.}$ Define the meet of two ideals $I$ and $J$ in $X$ by $I\cap J$ and the join of $I$ and $J$ by $I+J\text{.}$ Prove that the set of ideals of $R$ is a lattice under these operations.

By drawing the appropriate diagrams, complete the proof of Theorem 19.31 to show that the switching functions form a Boolean algebra.

Let $X$ and $Y$ be posets. A map $\varphi :X\to Y$ is order-preserving if $a⪯b$ implies that $\varphi (a)⪯\varphi (b)\text{.}$ Let $L$ and $M$ be lattices. A map $\psi :L\to M$ is a lattice homomorphism if $\psi (a\vee b)=\psi (a)\vee \psi (b)$ and $\psi (a\wedge b)=\psi (a)\wedge \psi (b)\text{.}$ Show that every lattice homomorphism is order-preserving, but that it is not the case that every order-preserving homomorphism is a lattice homomorphism.

Let $B$ be a Boolean algebra. Prove that $a=b$ if and only if $(a\wedge b^{\prime })\vee (a^{\prime }\wedge b)=O$ for $a,b\in B\text{.}$

Let $B$ be a Boolean algebra. Prove that $a=O$ if and only if $(a\wedge b^{\prime })\vee (a^{\prime }\wedge b)=b$ for all $b\in B\text{.}$

Let $L$ and $M$ be lattices. Define an order relation on $L\times M$ by $(a,b)⪯(c,d)$ if $a⪯c$ and $b⪯d\text{.}$ Show that $L\times M$ is a lattice under this partial order.