In this blog post we will explore the consequences of postulating 0 = 1 in an algebraic structure with two binary operations (S, +, 0) and (S, ⋅, 1). Such united monoids have a few interesting properties, which are not immediately obvious. For example, we will see that the axiom 0 = 1 is equivalent to a seemingly less extravagant axiom ab = ab + a, which will send us tumbling down the rabbit hole of algebraic graphs and topology.