Let \(X\) and \(Y\) be finite sets. A relation \(\alpha\) between \(X\) and \(Y\) is any subset of \(X\times Y\) and, when convenient, it will be identified with a Boolean matrix in the usual manner. A relation between \(X\) and \(X\) will be referred to as a relation on \(X\). The union (or sum) \(\alpha\cup\beta\) of two relations \(\alpha\) and \(\beta\) from \(X\) to \(Y\) is an unambiguous sum if \(\alpha\) and \(\beta\) are disjoint. If \(\alpha\) is a relation between \(X\) and \(Y\) and \(\beta\) is a relation between \(Y\) and \(Z\) then the product \(\alpha\beta\) is an unambiguous product if for each \((i,j)\in\alpha\beta\), there exists a unique \(k\) such that \((i,k)\in\alpha\) and \((k,j)\in\beta\). A semigroup \(S\) of relations on a set \(X\) is unambiguous if all products are unambiguous and it is transitive if for all \(i,j\in X\), \((i,j)\in\alpha\) for some \(\alpha\in S\). A decomposition of a relation \(\sigma\) between \(X\) and \(Y\) is a factorization \(\sigma=\alpha\beta\) where \(\alpha\) is a relation between \(X\) and \(Z\) and \(\beta\) is a relation between \(Z\) and \(Y\). The number \(|Z|\) is referred to as the size of the decomposition. If \(\alpha\beta\) is an unambiguous product, then the factorization is referred to as an unambiguous decomposition. The rank of a relation \(\sigma\) between \(X\) and \(Y\) is the minimal size of all its decompositions and is denoted by \(\rho(\sigma)\). The unambiguous rank is the minimal size of all its unambiguous decompositions and it is denoted by \(\rho_{\text{NA}}(\sigma)\). The author shows that the rank and unambiguous rank need not be equal for relations in a transitive unambiguous monoid. However, \(\rho(\sigma)\) and \(\rho_{\text{NA}}(\sigma)\) will coincide whenever \(\sigma\) is a regular element of an unambiguous semigroup.