Protractor: Finding the right element to test

I think I'm misunderstanding how elements works..

HTML code:

<div id="div-item">
    <a href="#">A link</a>
    <form>
        <div>
            <select>
                <option>1</option>
                <option>2</option>
            </select>
        </div>
    </form>
</div>

When I do this:

element(by.tagName('select')).all(by.tagName('option')).count();

This gives me 2, which is correct

When I do this:

element(by.id('div-item')).element(by.tagName('select')).all(by.tagName('option')).count();

This gives me 0. I thought chaining elements finds sub-elements. Is this not correct? How do I restrict the .all(by.tagName('option')) only within this div, rather than the whole page?