Design patterns are rarely used on their own. They are almost always to be found composed with each other in real applications. So it is crucial that we can reason about their compositions. In our previous work, we defined a set of operators on patterns so that pattern compositions can be represented as expressions on patterns. In this paper, we investigate the algebraic properties of these operators, prove a set of algebraic laws that they obey, and use the laws to show the equivalence of pattern compositions.