Abian's poset and the ordered monoid of annihilator classes in a reduced commutative ring

Let R be a reduced commutative ring with 1 not equal 0. Then R is a partially ordered set under the Abian order defined by x <= y if and only if xy = x(2). Let RE be the set of equivalence classes for the equivalence relation on R given by x similar to y if and only if ann(R)(x) = ann(R)(y). Then RE is a commutative Boolean monoid with multiplication [x][y] = [xy] and is thus partially ordered by [x] <= [y] if and only if [xy] = [x]. In this paper, we study R and R-E as both monoids and partially ordered sets. We are particularly interested in when R-E can be embedded in R as either a monoid or a partially ordered set.