Appendix A: Mathematical Preliminaries Definition A1. Cartesian Product Let X and Y be two non-empty sets. The Cartesian product of X and Y, denoted X  Y, is the set of all ordered pairs, a member of X followed by a member of Y. That is, ÈÉ X  Y ¼ ðÞx, y x 2 X, y ∈ Y : Example A2. Let X ¼ {0, 1}, Y ¼ {x, y, z}. Then the Cartesian product X  Y of X and Y is fgðÞ0, x ,0,ðÞy ,0,ðÞz ,1,ðÞx ,1,ðÞy ,1,ðÞz : Definition A3. Relation Let X and Y be two non-empty sets. A subset of X  Y is a relation from X to Y; a subset of X  X is a relation in X. Example A4. Let X ¼ {0, 1}, Y ¼ {x, y, z}. Then ÈÉÀ Á À Á À Á R ¼ ðÞ0, x , 1, x , 1, y , 1, z , R0 ¼ ∅, are relations from X to Y.(R0 is a relation because the empty set ∅ is a subset of every set.) Moreover, R00 ¼ fgðÞ0, 0 ,1,0ðÞ,1,1ðÞ is a relation in X. Definition A5. Function Let X and Y be two non-empty sets. E. Agmon, The Languages of Western Tonality, Computational Music Science, 265 DOI 10.1007/978-3-642-39587-1, © Springer-Verlag Berlin Heidelberg 2013 266 Appendix A: Mathematical Preliminaries A function f from X into Y is a subset of X  Y (i.e., a relation from X to Y) with the following property. For every x ∈ X, there exists exactly one y ∈ Y, such that (x, y) ∈ f. The set X is called the domain of f. The set of all y ∈ Y such that, for some x ∈ X,(x, y) ∈ f, is called the range of f, and is denoted RðÞf . For every (x, y) ∈ f we call y the value of f at x, and write: fðÞ¼x y: We sometimes write f : X ! Y to denote the function f from X into Y. We also call f: X ! Y a mapping of X into Y. Example A6. Let X ¼ {0, 1}, Y ¼ {x, y, z}. Then f ¼ fgðÞ0, x ,1,ðÞx is a function from X into Y satisfying f(0) ¼ f(1) ¼ x, RðÞ¼f fgx . Example A7. Let X ¼ {x, y, z}, Y ¼ {0, 1}. Then g ¼ fgðÞx,0, ðÞy,0, ðÞz,1 is a mapping of X into Y satisfying g(x) ¼ g( y) ¼ 0, g(z) ¼ 1. Note that the range of g in this case, unlike the range of f in Example A6, is Y. Definition A8. Surjection Let f : X ! Y be a function. We shall refer to f as a function from X onto Y,ora surjection,ifRðÞ¼f Y. Definition A9. Injection (One-to-One Function) Let f : X ! Y be a function. We shall say that f is a one-to-one function from X into Y,oran injection,if f(x) ¼ f(x0) implies that x ¼ x0 for every x, x0 ∈ X. Example A10. Let X ¼ {x, y, z}, Y ¼ {0, 1, 2, 3}. Then h ¼ fgðÞx,2, ðÞy,3, ðÞz,1 is a one-to-one mapping of X into Y, or an injection, since h(x) ¼6 h(y) 6¼ h(z) ¼6 h(x). Definition A11. Bijection Let f : X ! Y be a function. If f is both a surjection and an injection we shall say that f is a one-to-one function from X onto Y,ora bijection. Appendix A: Mathematical Preliminaries 267 Example A12. Let X ¼ {x, y, z}, Y ¼ {0, 1, 2}. Then i ¼ fgðÞx,2, ðÞy,0, ðÞz,1 is a one-to-one function from X onto Y, or a bijection. We have both i(x) 6¼ i( y) 6¼ i(z) 6¼ i(x) and RðÞ¼i fgiðÞx ,iðÞy ,iðÞz ¼ Y. Definition A13. Induced Function Let X and Y be two non-empty sets, and let f be a function from X into Y. Let X0 be a non-empty subset of X, and let f0 be a function from X0 into Y. We shall say that f0 is induced by f, if, for every x0 ∈ X0,f0(x0) ¼ f(x0). If f0 is induced by f, we shall refer to f0 as the restriction of f to the set X0. Example A14. Let h : X ! Y be as in Example A10, and let X0 ¼ {x, z} X ¼ {x, y, z}. Then the function h0 ¼ fgðÞx,2, ðÞz,1 is induced by h. Definition A15. Semigroup Let S be a non-empty set, and let È be an operation on S, that is, a mapping of S  S into S. We shall refer to S ¼ {S, È}asa semigroup if È is associative. That is, for every s, t, and u in S,(s È t) È u ¼ s È (t È u). Definition A16. Group; Abelian Group Let S ¼ {S, È} be a semigroup. We shall refer to S as a group if S has an identity element e, and every element s in S has an inverse –s. In other words, for every element s in S there exists: A. An element e such that s È e ¼ s ¼ e È s. B. An element –s such that s ÈÀs ¼ e ¼s È s. If S is commutative, that is, s È t ¼ t È s for every s, t in S, we shall refer to S as “abelian.” Example A17. Let be the set of real numbers and let + be the usual operation of addition on the reals. Then fg, þ is a group. The identity element of the group is 0, and the inverse of every element x in is –x. # \ Let Èɼ fg0 and let Á be the usual operation of multiplication on the reals. # Then , Á is a group. The identity element of the group is 1, and the inverse of every element x in # is 1/x. 268 Appendix A: Mathematical Preliminaries Definition A18. Homomorphism Let S ¼ {S, È} and S0 ¼ {S0, } be two semigroups, and let f be a mapping of S into S0. We shall refer to f as a homomorphism of S into S0 if fðÞ¼s È t fðÞ s fðÞt for every s, t ∈ S. Example A19. Let S ¼ fg, þ be the group of all real numbers under addition, and let S0 ¼ fgþ, Á be the group of all strictly positive real numbers under multiplication. Fix a strictly positive number B, and let f(x) ¼ Bx for every x in . Then f is a homomorphism of S into S0. We have: fðÞ¼x þ y Bxþy ¼ Bx Á By ¼ fðÞÁx fðÞy for every x, y ∈ . Appendix B: Modules and Their Homomorphisms Definition B1. Ring Let R be a non-empty set, and let + and Á be two operations on R. We shall refer to R ¼ {R,+,Á}asa ring if {R, +} is an abelian group, {R, Á}is a semigroup, and the following two conditions, A and B, are satisfied for every α, β, and γ in R. A. γ Á ðÞ¼α þ β ðÞþγ Á α ðÞγ Á β : B. ðÞÁα þ β γ ¼ ðÞþα Á γ ðÞβ Á γ : If the semigroup {R, Á} has an identity element we shall refer to R as a ring with identity; if the operation Á is commutative we shall refer to R as a commutative ring. Example B2. Let + and Á be the usual operations of addition and multiplication on the reals. Then R ¼ {R,+,Á} is a commutative ring with identity if A. R ¼ is the set of integers. B. R ¼ is the set of rational numbers. C. R ¼ is the set of real numbers. Definition B3. Module Let R ¼ {R,+,Á} be a ring with identity 1, let S ¼ {S, È} be an abelian group, and let • be a function from R  S into S. Write •(α, s) ¼ α • s. We shall refer to S ¼ {R, S,•}asa module over R if 1 • s ¼ s for every s ∈ S, and the following three conditions, A, B, and C, are satisfied for every α, β ∈ R, s, t ∈ S. E. Agmon, The Languages of Western Tonality, Computational Music Science, 269 DOI 10.1007/978-3-642-39587-1, © Springer-Verlag Berlin Heidelberg 2013 270 Appendix B: Modules and Their Homomorphisms A. ðÞα þ β s ¼ ðÞÈα s ðÞβ s : B. α ðÞ¼s È t ðÞÈα Á s ðÞα t : C. α ðÞ¼β s ðÞα Á β s: The following proposition, which may be checked easily against Definition B3, states that any abelian group may be viewed as a module over the ring of integers in a unique way. Proposition B4. Let {R,+,Á} be the ring of integers, let {S, È} be an abelian group with identity e, and let • be a function from  S into S. Then S ¼ {R, S, •} is a module if, and only if, 0 • s ¼ e, and the following two conditions, A and B, are satisfied for every natural n and every s in S. n times zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ A. n s ¼ s È ::: È s : B. ðÞÀn s ¼ðÞn s : We shall refer henceforth to a module over the ring of integers as a module. A homomorphism from group S into group S0 is a mapping from S into S0 that preserves the group “addition” (see Appendix A). If S and S0 are two modules over the same ring R, a homomorphism from S into S0 is a group homomorphism from S into S0 that preserves the module “multiplication.” Definition B5.