Algebraic Properties

These sections include the algebraic “closure" properties of distributive and commutative functions or operators. We will follow Servois in calling them all functions, even though in many cases the objects would be called operators today.

In Section 5, Servois proves that distributivity is closed under composition: if two functions \( \varphi \) and \(\psi\) are distributive, then the composition of \(\varphi\) and \(\psi\) is also distributive. It follows immediately that different orders of distributive functions are also distributive. In Section 6, Servois shows that distributivity is closed under addition.

Next, Servois tackles the properties of commutative operations. In Section 7, he shows that given a collection of \(n > 2\) functions that are pairwise commutative, all of the \(n!\) composed functions that can be formed from them are equal.  Servois gives an example of such a situation: if \(\mbox{f}, f,\) and \(\varphi\) are three operators, which are commutative between themselves, then we have \[\mbox{f}(f(\varphi (z))) = f(\mbox{f}(\varphi (z))) = \mbox{f}(\varphi (f(z))) = \varphi (\mbox{f}(f(z))) = f (\varphi (\mbox{f}(z))) = \varphi (f(\mbox{f}(z))).\] Servois provides a proof by induction for this theorem. In these sections, we sometimes witness Servois laboring over intricate details of somewhat “obvious” theorems, illustrating his devotion to rigor.

In Section 8, Servois proves that if \({\mbox f}\) and \(f\) are commutative between themselves, then they are also commutative with their inverses. In Section 9 he deduces from Sections 7-8 that, given a set of commutative functions, the positive integral powers of these functions also commute amongst themselves.

In Sections 10-12, Servois considers collections of distributive functions that commute with each other. He shows that a sum \(F=\sum_i f_i\) of such functions commutes with all of the constituent functions \(f_i\), as do the powers \(F^n\). As a consequence, two such functions \(F=\sum_i f_i\) and \(G=\sum_i g_i\) also commute. Servois does not use subscript and summation notation; his proofs might be easier for a modern reader to follow if he did. We also note that Servois seems to be including the case of infinite sums, although he does not say so explicitly. However, he will eventually define the differential operator \({\mbox d}\) as an infinite sum of distributive operators, so it is clear that he means to include the case of infinite sums.

In Sections 5-12, Servois essentially shows that the class of invertible, distributive, and pairwise commutative functions forms a field with respect to the operations of addition and composition. He does not prove that these operations satisfy the associative laws, but it wasn’t until later in the nineteenth century that mathematicians paid attention to the associative property, although Carl Friedrich Gauss (1777-1855) did prove an associative law in [Gauss 1801, Section 240]. In addition, he assumes that all his functions are invertible, rather than proving the existence of inverses. It is important to bear in mind, however, that Servois wasn’t interested in abstract algebraic structures, although he influenced the British school of symbolical algebra [Allaire and Bradley 2002].