Problem 9 - Applied Abstract Algebra

Question

An isomorphism of posets is a bijective order-homomorphism, whose inverse is also an order-homomorphism. Prove: If \(f\) is an isomorphism of a poset \(L\) onto a poset \(M\), and if \(L\) is a lattice, then \(M\) is also a lattice, and \(f\) is an isomorphism of the lattices.

Step-by-Step Solution

Answer

'f' not only is an isomorphism between 'L' and 'M' as posets, also if 'L' is a lattice, 'M' also becomes a lattice with this isomorphism maintained. Hence, 'f' becomes an isomorphism between the two lattices 'L' and 'M'.

Step 1: Define Relevant Terms

A poset involves two elements, let's say a and b, with an order relation between them. A lattice is a special type of poset where any two elements have both a unique sup (least upper bound) and a unique inf (greatest lower bound). An isomorphism is a structure-preserving map between two algebraic structures. In our context, we talk about order-isomorphisms, which preserve the order of elements.

Step 2: Apply the Definition of an Isomorphism

Given that 'f' is an isomorphism between posets 'L' and 'M', it means that 'f' is bijective and both 'f' and its inverse 'f⁻¹' are order-preserving. This means for all a and b in L, if a is less than or equal to b in L, then 'f(a)' is less than or equal to 'f(b)' in M. And conversely, if 'c' is less than or equal to 'd' in M, with c = f(a) and d = f(b) for some a, b in L, then a is less than or equal to b.

Step 3: Apply the Definition of a Lattice to 'L'

L is given as a lattice, so for all 'a' and 'b' in 'L', there exist least upper bound (sup) say 'c' and greatest lower bound (inf) say 'd' in 'L' such that c = sup(a, b) and d = inf(a, b).

Step 4: Show that 'M' is a Lattice

We now go on to show that 'M' is also a lattice. Given 'a' and 'b' in 'L', and 'f(a)' and 'f(b)' in 'M' because 'f' is bijective. The least upper bound of 'f(a)' and 'f(b)' in 'M', denoted as sup_M(f(a), f(b)), will be 'f(c)'. Similarly, the greatest lower bound of 'f(a)' and 'f(b)' in 'M', denoted as inf_M(f(a), f(b) will be 'f(d)'. This shows that 'M' is also a lattice.

Step 5: Prove 'f' is an Isomorphism of Lattices

Given that we've shown 'f' is an isomorphism of posets and that if 'L' is a lattice so is 'M', now we can say 'f' is also an isomorphism of lattices. This is because it has preserved the lattice structure from 'L' to 'M'.

Key Concepts

Lattice Theory

Lattice theory studies mathematical structures that allow ordering of their elements in a consistent way, where it's always possible to find a common ground between any two elements. These structures, called lattices, have both an algebraic and a geometric interpretation, which makes them useful in various fields, such as abstract algebra and computer science.

In the context of our exercise, a lattice is defined as a poset (partially ordered set) in which any two elements have a least upper bound—called the 'sup' (supremum)—and a greatest lower bound—termed the 'inf' (infimum). For a simple example, consider the set of integers under the usual 'less than or equal to' relation; here, the 'sup' and 'inf' of any two integers can be thought of as their least common multiple and greatest common divisor, respectively.

Understanding lattice theory is crucial because it provides the underlying framework for discussing poset isomorphism and the preservation of structure in bijective order-homomorphisms, as seen in the exercise.

Bijective Order-Homomorphism

When we talk about bijective order-homomorphism, we delve into a concept that is fundamental to the notion of structure preservation in order theory. An order-homomorphism is a function between two posets that maintains the ordering of elements. This means if you have two elements in the first poset, and one is less than the other, their images under this function in the second poset will preserve this order relationship.

Bijection is another important property for isomorphisms, which means that the function is both injective (never maps distinct elements to the same point) and surjective (covers every point in the target set). Put these two properties together, and you have a bijective order-homomorphism: a function that is a perfect 'matchmaker' between two posets, ensuring every element has a unique partner and their relational structure is maintained.

Algebraic Structures

Algebraic structures are sets equipped with one or more operations that follow certain rules. These operations take elements from the set and return new elements. Examples include groups, rings, and fields, each having distinct properties and axioms. A poset itself is a simple example of an algebraic structure where the 'operation' is less about numerical computation and more about the comparison of elements.

The exercise focuses on the intricacies of mapping between these structures. An isomorphism in lattice theory is a special kind of bijective order-homomorphism that not only maps one poset onto another but also ensures that the structure defined by the suprema and infima is preserved. So, it's not just about maintaining a list of relationships between elements; it's about ensuring that the fundamental nature of the operations within the structure—like finding 'sup' and 'inf'—remains consistent after mapping. This affirms that the structures' operational integrity is maintained.