Adding setters to properties in overrides

Why is it allowed to change the visibility and existence of getters or setters in a property when implementing an interface?

interface IFoo
{
    string Bar { get; }
}

class RealFoo : IFoo
{
    public RealFoo(string bar)
    {
        this.Bar = bar;
    }

    public string Bar { get; private set; }
}

class StubFoo : IFoo
{
    public string Bar { get; set; }
}

...and not legal to do the same when implementing an abstract class?

abstract class AbstractFoo : IFoo
{
    public abstract string Bar { get; }
}

class RealFoo : AbstractFoo
{
    public RealFoo(string bar)
    {
        this.Bar = bar;
    }

    // Cannot override because 'Bar' does not have an overridable set accessor
    public override string Bar { get; private set; }
}