Question

Say for each of the posets represented by the given Hasse-diagram whether the poset is

i) a lattice

ii) a complemented lattice

iii) a Boolean algebra

Give reasons for your answers

Answer 1

Solution:

Here Hasse diagram is missing, so we will define the given terms:

(a) A poset is short for partially ordered set which is a set whose elements are ordered but not all pairs of elements are required to comparable in the order.

(i) A lattice is a poset (𝑋, 𝑅) with two properties: • 𝑋 has an upper bound 1 and a lower bound 0; • for any two elements 𝑥, 𝑦 ∈ 𝑋, there is a least upper bound and a greatest lower bound of a set {𝑥, 𝑦}. 

(ii) A complemented lattice is an algebraic structure $\left(L, \wedge, v, 0,1,^{\prime}\right)$ such that $(L, \wedge, v, 0,1)$ is a bounded lattice and for each element $x \in L$ , the element $x^{\prime} \in L$ is a complement of x, meaning that it satisfies

1. $x \wedge x^{\prime}=0$

2. $x \vee x^{\prime}=1$

(iii) A Boolean algebra (BA) is a set A together with binary operations + and $\cdot$ and a unary operation -, and elements 0,1 of A such that the following laws hold: commutative and associative laws for addition and multiplication, distributive laws both for multiplication over addition and for addition over multiplication, and the following special laws:

$\begin{array}{r} (x+(x \cdot y)=x \\ (x \cdot(x+y)=x \\ x+(-x)=1 \\ x \cdot(-x)=0 \end{array}$