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P1: FVZ Revised Pages Qu: 00, 00, 00, 00 Encyclopedia of Physical Science and Technology EN001C.19 May 26, 2001 14:19 Algebra, Abstract Ki Hang Kim Fred W. Roush Alabama State University I. Sets and Relations II. Semigroups III. Groups IV. Vector Spaces V. Rings VI. Fields VII. Other Algebraic Structures GLOSSARY other structure T such that, for every operation ∗ and all x, y in S, f (x ∗ y) = f (x) ∗ f (y). Binary relation Set of ordered pairs (a, b), where a, b Ideal Subset ℑ of a ring containing the sum and differ- belong to given sets A, B; as the set of ordered pairs of ence of any two elements of ℑ, and the product of any real numbers (x, y) such that x > y. element of ℑ with any element of the ring. Equivalence relation Binary relation on a set S often Identity Element e such that, for all x for a given opera- written x ∼ y if (x, y) is in the relation such that (1) tion ∗, e ∗ x = x ∗ e = x. x ∼ x; (2) if x ∼ y then y ∼ x; and (3) if x ∼ y and Incline Structure having two associative and commuta- y ∼ z then x ∼ z for all x, y, z in S. tive operations + and ×, satisfying the distributive law Field Structure having operations of commutative, as- and (1) x + x = x and (2) x + (x × y) = x. sociative distributive addition and multiplication with Isomorphism Homomorphism f from a set S to a set additive and multiplicative identities and inverses of T such that the inverse mapping g from T to S nonzero elements, as the real or rational numbers. where g(y) = x if and only if f (x) = y is also a Function Binary relation from a set A to a set B that to homomorphism. every a in A associates a unique b in B such that (a, b) Partial order Binary relation R on a set S such that for is in the (binary) relation, as a polynomial associates all x, y, z in S (1) (x, x) is in R; (2) if (x, y) and (y, x) its value to a given x. are both in R, then x = y; and (3) if (x, y) and (y, z) Group Structure having one associative operation with are in R, so is (x, z). identity and inverses. Quadratic form A function defined on a ring ᏾ by Homomorphism Function f from one structure S to an- ai j xi x j . 435 P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 436 Algebra, Abstract Quotient group For a normal subgroup N of a group G, ments in B but not in A. For fixed set B ⊃ A, this is called the group formed by equivalence classes x¯ under the the complement of A in B. −1 relation that x ∼ y if xy in N, with multiplication The Cartesian product A1 × A2 × · · · × An of n sets given by x¯ y¯ = xy. A1, A2, . . . , An is the set of all n-tuples (a1, a2, . . . , an) Ring Structure having one associative and commutative such that ai ∈ Ai for all i = 1, 2, . . . , n. Infinite Cartesian operation with identity and inverses, and a second op- products can be similarly defined with i ranging over any eration associative and distributive with the first. index set I. Semigroup Structure having an associative operation. Simple Structure S such that, for every homomorphism B. Binary Relations f to another structure, either f (x) = f (y) for all x, y in S or f (x) ≠ f (y) for all x ≠ y in S. The truth or falsity of any mathematical statement about 2 2 Subgroup Subset S of a group G containing the product a relationship x R y, for instance, x + y = 1, can be de- and inverse of any of its elements. It is called normal termined from the set of ordered pairs (x, y) such that −1 if, for every g in G, s in S, the element gsg is in S. x R y. In fact, R is equivalent to the relationship that (x, y) ∈ {(x, y) ∈ A × B : x R y}, where A and B are sets containing x and y. ABSTRACT ALGEBRA is the study of laws of opera- The set of ordered pairs then represents the relationship tions and relations such as those valid for the real numbers and is called a binary relation. In general, a binary relation and similar laws for new systems. It studies the class of from A to B is a subset of A × B, that is, a set of ordered all possible systems having given laws, for example, asso- pairs. ciativity. There is no standard object that it studies such as The union, intersection, and complement of binary re- the real numbers in analysis. Two systems are isomorphic lations from A to B are defined on them as subsets of in abstract algebra if there is a one-to-one (1–1 for short) A × B. Another operation is composition. If R is a binary correspondence between their elements such that relations relation from A to B and S is a binary relation from B to and operations agree in the two systems. C, then R ◦ S is {(x, z) ∈ A × C : (x, y) ∈ R and (y, z) ∈ S for some y ∈ B}. Composition is distributive over union. For R, R1, S, S1, T , from A to B, A to B, B to C, B to C, C to D, we have (R ∪ R1) ◦ S = (R ◦ S) ∪ (R1 ◦ S) and I. SETS AND RELATIONS R ◦ (S ∪ S1) = (R ◦ S) ∪ (R ◦ S1). It is associative; we have (x, w) ∈ (R ◦ S) ◦ T if and only if for some y ∈ B and z ∈ C, A. Sets (x, y) ∈ R, (y, z) ∈ S, and (z, w) ∈ T . The same condition A set is any precisely defined collection of objects. The is obtained for R ◦ (S ◦ T ). objects in a set are called elements (or members) of the The identity relation 1A is {(a, a) : a ∈ A}. This rela- set. The set A = {1, 2, x, y} has elements 1, 2, x, y and we tion acts as an identity on either side for R ⊂ A × B, write 1 ∈ A, 2 ∈ A, x ∈ A, and y ∈ A. There is no set of all S ⊂ B × C; that is, R ◦ 1B = R, and 1B ◦ S = S. T sets, but given any set we can obtain other sets by oper- The transpose R of a binary relation R from A to B is ations listed below, and for any set A and property P(x) {(b, a) ∈ B × A : (a, b) ∈ R}. It is also called converse and we have a set {x ∈ A : P(x)} of all elements of A having inverse. the property. Things such as “all sets” are called classes rather than sets. There is a set of all real numbers and one C. Functions of all isosceles triangles in three-dimensional space. We say that A is a subset of B if all elements of A are A partial function from a set A to a set B is a relation f in B. This is written A ⊂ B, or B ⊃ A. The set is called a from A to B such that if (x, y) ∈ f , (x, z) ∈ f then y = z. If proper subset if A ≠ B; otherwise it is called an improper (x, y) ∈ f , we write y = f (x) since this condition means subset. For finite sets, a subset is proper if and only if it y is unique. A partial function is a function defined on a has strictly fewer elements. The empty set ∅ is the set with subset of a set considered. Its domain is {a ∈ A : (a, b) ∈ f no elements. for some b ∈ B}. If A ⊂ B and B ⊂A, then A = B. The union ∪ᑠ A of a A function is a partial function such that for all x ∈ A family ᑠ of sets is the set whose elements are all things there exists y ∈ B with (x, y) ∈ f . that belong to at least one set A ∈ ᑠ. The intersection ∩ᑠ A A function is 1–1 if and only if whenever (x, z) ∈ f , is the set of all elements lying in every set A in ᑠ. (y, z) ∈ f we have x = y. This is the transpose of the def- The power set P(A) is the set of all subsets of A. The inition of a partial function. Functions and partial func- relative complement B\A (or B − A) is the set of all ele- tions are thought of in several ways. A function may be P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 Algebra, Abstract 437 considered the output of a process in which x is an input, The set x¯ = {y ∈ A : x R y} is called the equivalence 2 as x + x + 1 adds a number to its square and 1. We may class of x (the set of all elements equivalent to the same consider it as assigning an element of B to any element of element x). The set of equivalence classes x¯ is called A. Or we may consider it to be a map of A into the set B the quotient set A/R. The function f (x) = x¯ is called in which the point x ∈ A is represented by f (x) ∈ B. Then the quotient map A → A/R. The relation {(x, y) ∈ A × f (x) is called the image of x. Also for a subset C ⊂ A the A : f (x) = f (y)} is precisely R. image f (C) is { f (x) : x ∈ C} the set of images of points of Any two equivalence classes are disjoint or equal: If C. We may also consider it to be a transformation. Strictly, x R z and y R z, by symmetry z R y and by transitivity however, the term transformation is sometimes reserved x R y. Any element belongs in the equivalence class of for functions from a set to itself. itself. A family ᑠ of nonempty subsets of a set S is called a A 1–1 function sends distinct values of A to distinct partition if (1) whenever C ≠ D in ᑠ, we have C ∩ D = ∅; points. For a 1–1 function on a finite set C, f (C) will (2) ∪ᑠ C = S. Therefore, the set of equivalence classes have exactly as many elements as C. always forms a partition. Conversely, every partition ᑠ The composition of functions f : A → B and g : B → C arises from the equivalence relation {(x, y) ∈ A × A: for is given by g( f (x)). It is written g ◦ f . Composition is as- some C ∈ ᑠ, x ∈ C and y ∈ C}. sociative since it is a special case of composition of binary An important equivalence is congruence modulo m of relations except that the order is reversed. The identity re- integers. We say x ≡ y(mod m) for integers x, y, m if lation on a set is also called the identity function. A com- there exists an integer h such that x − y = hm, that is, position of partial (1–1) functions is respectively a partial if m divides x − y. If x − y = hm and y − z = km, then (1–1) function. x − z = x − y + y − z = hm + km. So if x ≡ y(mod m) and y ≡ z(mod m), then x ≡ z(mod m). This proves transitivity. A relation that is reflexive and transitive is called a quasiorder. If it satisfies also (OR-4) antisymmetry if A function f : A → B is onto if f (A) = B, that is, if (x, y) ∈ R and (y, x) ∈ R then x = y, it is called a par- every y ∈ B is the image of some x. A composition of tial order. If a partial order satisfies (OR-5) completeness onto functions is onto. A 1–1 onto function is called a 1–1 for all x, y ∈ A either (x, y) ∈ R or (y, x) ∈ R, it is called correspondence. For a 1–1 onto function f , the inverse a total order, or a linear order. (converse) of f is defined by f −1 = {(y, x) : (x, y) ∈ f } For every total order on a finite set A, the elements or x = f −1(y) if and only if y = f (x). It is character- of A can be labeled a1, a2, . . . , an in such a way that ized by f ◦ f −1 = 1B, f −1 ◦ f = 1A. Moreover, for any ai R a j if i < j. An isomorphism between a binary rela- binary relations R, S if R ◦ S = 1B, S ◦ R = 1A both R and tion R1 on a set A1 and R2 on A2 is a 1–1 correspon- S must be 1–1 correspondences and S must be RT. A dence f : A1 → A2 such that (x, y) ∈ R1 if and only if 1–1 correspondence from a finite set to itself is called a ( f (x), f (y)) ∈ R2. Therefore, every total order on a finite permutation. set is isomorphic to the standard order on {1, 2, . . . , n}. There are many nonisomorphic infinite total orders on the same set and many nonisomorphic partial orders on fi- D. Order Relations nite sets of n elements. The structure of quasiorders can be That a binary relation R from a set A to itself is (OR-1) re- reduced to that of partial orders. For every quasiorder R on flexive means (x, x) ∈ R for all x ∈ A; (OR-2) symmetric any set A, the relation {(x, y) : (x, y) ∈ R and (y, x) ∈ R} means if (x, y) ∈ R, then (y, x) ∈ R; and (OR-3) transitive is an equivalence relation. The quasiorder gives a partial means if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R. Fre- order on the set of equivalence classes of R and (x, y) ∈ R quently x R y is written for (x, y) ∈ R, so that transitivity if and only if (x¯ , y¯ ) ∈ R1. means if x R y and y R z, then x R z. For instance, if x ≥ y The structure of partial orders on small sets can be de- and y ≥ z, then x ≥ z. So x ≥ y is transitive, but x ≠ y is scribed by diagrams known as Hasse diagrams. An ele- not transitive. ment x of a partial order is called minimal (maximal) if The following relations are reflexive, symmetric, and for no y ≠ x does (y, x) ∈ R ((x, y) ∈ R) where (x, y) ∈ R transitive on the set of geometric figures: x = y (same is taken as x ≤ y in the order. Every partial order on a finite point set), x ≃ y (congruence), x ∼ y (similarity), x has set has at least one minimal and at least one maximal ele- the same area as y. A reflexive, symmetric, transitive re- ment. Represent all minimal elements of the partial order lation is called an equivalence relation. For any function as points at the same horizontal level of the bottom of the f from A to B the relation {(x, y) ∈ A × A : f (x) = f (y)} diagram. From then on, the i th level consists of elements is an equivalence relation. z not in previous levels but such that for at least one y on P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 438 Algebra, Abstract E. Boolean Matrices and Graphs The Boolean algebra has elements {0, 1} and operations c +, ·, given by 0+0=0, 0+1=1+0=1+1=1, 0 ·0= c c 1 · 0 = 0 · 1= 0, 1 · 1 = 1, 0 = 1, 1 = 0. There are many interpretations and uses of the Boolean algebra : (1) propositional logic where 0 is false, 1 is true, + is “or,” · FIGURE 1 Hasse diagrams of partially ordered sets. c is “and,” is “not,” (2) switching circuits where 0 means no current flows, 1 means current flows; and (3) × × · · · × can be taken as the set of subsets of an n-element a previous level y < z and for no x is y < x < z. That is, z is the very next element greater than y. For all such z , y set {yi }, where (x1, x2, . . . , xn) corresponds to the subset draw a line segment from z to y. Figure 1 gives the Hasse {yi : xi = 1}; (4) 0 means zero in R, 1 means some positive number in R, where R denotes the set of all real numbers. diagrams of three partially ordered sets (posets for short). The algebra satisfies the same laws as the algebra of A poset is called a lattice if every pair of elements in sets under ∪, ∩, ∼ where ∼ denotes the complementation it have a least upper bound and a greatest lower bound. since it is a Boolean algebra. An upper (lower) bound on a set S is an element x such that y ≥ x (y ≤ x) for all y ∈ S. Every linearly ordered set An n × m Boolean matrix A = (ai j ) is an n × m rect- angle of elements of . The entry in row (horizontal) is a lattice, and any family of subsets of a set containing the union and intersection of any two of its mem- i and column (vertical) j is denoted ai j . To every bi- bers. These lattices are distributive: If ∧, ∨ denote great- nary relation R from a set X = {x1, x2, . . . , xn} to set est lower bound, least upper bound, respectively, then Y = {y1, y2, . . . , ym} we can assign a Boolean matrix M = x ∧(y ∨ z)=(x ∧ y)∨(x ∧ z) and x ∨ (y ∧ z)=(x ∨ y) ∧ (mi j ), where mi j is 0 or 1 according to whether (xi , y j ) does or does not lie in R. This gives a 1–1 onto correspon- (x ∨ z). A lattice is called modular if these laws hold dence from binary relations from X to Y to n × m Boolean whenever two of x, y, z are comparable [two elements matrices. Many questions can be dealt with simply in the a, b are comparable for R if (a, b) ∈ R or (b, a) ∈ R]. Boolean matrix form. Boolean matrices are multiplied and There exist nonmodular lattices, as the last diagram of added by the same formulas as for ordinary matrices, ex- Fig. 1. cept that operations are Boolean. Composition (union) of A binary relation R is called a strict partial order if it is binary relations corresponds to product (sum) of Boolean transitive and (OR-6) irreflexive, for no x does (x, x) ∈ R. matrices. There is a 1–1 correspondence between partial orders A directed graph (digraph for short) consists of a set V R (as x ≤ y) on A and strict partial orders (x < y) ob- of elements called vertices (represented by points) and a tained as R\{(a, a) : a ∈ A}. An irreflexive binary relation set E or ordered pairs of V known as edges. Each ordered R on A is called a semiorder if for all x, y, z, w ∈ A, pair (x, y) is represented by a segment with an arrow going (1) if (x, y) ∈ R and (z, w) ∈ R, then either (x, w) ∈ R from point x to point y. Thus digraphs are essentially the or (z, y) ∈ R; (2) if (x, y) ∈ R and (y, z) ∈ R, then either same as binary relations from a set to itself. Any n × n (w, z) ∈ R or (x, w) ∈ R. Semiorders can be represented (n-square) Boolean matrix or binary relation on a la- always as {(x, y) : f (x) > f (y) + δ} for some real valued beled set can be represented as a digraph, and conversely. function f and number δ, and are strict partial orders. Figure 3 shows the Boolean matrix and graph of a binary Figure 2 shows the relationship of many of the relations relation on {1, 2, 3, 4}. taken up here. F. Blockmodels Often binary relations are empirically obtained. Such binary relations can frequently be simplified by blocking the Boolean matrices: dividing the set of indices into disjoint subsets, relabeling to get members of the same subset adjacent, and dividing the matrix into blocks. Each nonzero block is replaced by a single 1 entry and each zero block by a single 0 entry. Many techniques called clustering exist for dividing the total set into subsets. One (CONCOR) is to take iterated correlation matrices of the FIGURE 2 Classification of binary operations. rows. P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 Algebra, Abstract 439 FIGURE 3 Matrix and graph of a relation. Provided that every nonzero block has at least one 1 in II. SEMIGROUPS each row the replacement of blocks by single entries will preserve all Boolean sums and products. A. Generators and Relations A binary operation is a function that given two entries from a set S produces some element of a set T . Therefore, G. General Relational Structures it is a function from the set S × S of ordered pairs (a, b) A general finite relational structure on a set S is an indexed to T . The value is frequently denoted multiplicatively as family Rα of subsets of S ∪ (S × S) ∪ (S × S × S) ∪ · · ·. a ∗ b, a ◦ b, or ab. Addition, subtraction, multiplication, Such structures include order structures and operational and division are binary operations. structures such as multiplicative ones as subsets {(x, y, z) The set S is said to be closed under the operation if the ∈ S × S × S : x ∗ y = z}. A homomorphism of relational product always lies in S itself. The positive integers are structures (index the same way) Rα on S1 to Tα on S2 con- not closed under subtraction or division. sists of a function f : S1 → S2 such that if g is the mapping The operation is called associative if we always have S1 ∪ (S1 × S1) ∪ (S1 × S1 × S1) ∪ · · · to S2 ∪ (S2 × S2) ∪ (a ◦ b) ◦ c = a ◦ (b ◦ c). We have noted that this always (S2 × S2 × S2) ∪ · · · which is f on each coordinate then holds for composition of functions or binary relations. g(Rα) ⊂ Tα for each α. An isomorphism of relational Conversely, if closure and associatively hold, the set can structures is a 1–1 onto homomorphism such that g(Rα) = always be represented by a set of functions under compo- Tα for each α. The quotient structure of Rα on S associ- sition. ated with an equivalence relation E on S is the structure A set with a binary operation satisfying associativity Tα = g(Rα), for f the mapping S → S/E assigning to each and closure is called a semigroup. The positive integers element its equivalence class. form a semigroup under multiplication or addition. An element is called a left (right) identity in a semigroup S if for all x ∈ S, x = x(x = x). A semigroup with two- H. Arithmetic of Residue Classes sided identity is called a monoid. To represent a semigroup as a set of functions under composition, first add a two- Let Z denote the set of all integers. Let Em be the sided identity element to obtain a monoid M. Then for equivalence relation {(x, y) : Z ×Z : x − y = km for some each x in M define a function fx : M → M by fx (y) = k ∈Z}. This is denoted x ≡ y(mod m). We have previously x ◦ y. noted that it is an equivalence relation. The set generated by a subset G ⊂ S is the set of all It divides the integers into exactly m equivalence + + finite products {x1x2 · · · xn : n ∈ Z , xi ∈ G}, where Z ¯ classes, for m ≠ 0. For m = 3, the classes are 0 = {. . . , denotes the set of all positive integers. The set satisfies ¯ −9, −6, −3, 0, 3, 6, 9, . . .}, 1 = {. . . , −8, −5, −2, 1, 4, ¯ 7, 10, . . .}, 2 = {. . . , −7, −4, −1, 2, 5, 8, 11, . . .}. Any two members of the same class are equivalent (3 divides their difference). This relation has the property that if x ≡ y(mod m) then, for any z ∈ Z, x + z ≡ y + z since m divides x + z − (y + z) = x − y and xz ≡ yz. Such a relation in general is called a congruence. For any congruence, we can define operations on the classes by x + y = x¯ + y¯ , and xy = x¯ y¯ . Let Zm be the set of equivalence classes of Z under congruence module m and under +, × quotient operators. FIGURE 4 Addition and multiplication of module 4 residue Operations in Z5 are given in Fig. 4. classes. P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 440 Algebra, Abstract closure since (x1x2 · · · xn)(y1y2 · · · yn) has agian the form B. Green’s Relations required. A subset of a semigroup satisfying closure is it- For analyzing the structure of semigroups in many cases self a semigroup and is called a subsemigroup. The set G it is best to decompose it into a family of equivalence is called a set of generators for S if G generates the entire classes. semigroup. Let S be a semigroup and let M be S with an identity el- For a given set G of generators of S, a relation is an ement if S lacks one. Let ᏾ be the relation for x, y ∈ S that equation x1x2 · · · xn = y1y2 · · · yn, where xi , yi ∈ G. for some a, b ∈ M, xa = y, yb = x. Let ᏸ be the relation A homomorphism of semigroups is a mapping f : for some a, b ∈ M, ax = y, by = x. Let ᏶ be the relation S1 → S2 such that f (x ◦ y) = f (x) f (y) for all x, y ∈ S1. z that for some a, b, c, d ∈ M, axc = y, byd = x. What these For example, e is a homomorphism from the additive equations express is a kind of divisibility, that either x, y semigroup of real numbers (R, +) to the multiplicative + x+y can be a factor of the other. For example, in Z under semigroup of complex numbers (C, ×), since e = x y multiplication (2, 5) ∉ ᏶ because 2 is not a factor of 5. (e )(e ). Here C denotes the set of all complex numbers. There are two other important relations: Ᏼ = ᏸ ∩ ᏾ = The trace and determinant of n-square matrices are ho- {(x, y) : x ᏸ y and x ᏾ y} and Ᏸ = ᏸ ◦ ᏾ = ᏾ ◦ ᏸ. These momorphisms, respectively, from the additive and multi- are both equivalence relations also and are known as plicative semigroups of n-square matrices to the additive Green’s relations. and multiplicative semigroups of real numbers. For the semigroup of transformations on a finite set An isomorphism of semigroups is a 1–1 onto homomor- T, f ᏸg if and only if the partitions given by the equiv- phism. The inverse function will then also be an isomor- alence relations {(x, y) ∈ T × T : f (x) = f (y)}, {(x, y) ∈ phism. Isomorphic semigroups are structurally identical. T × T : g(x) = g(y)} coincide, f Ᏸ g if and only if f ᏶ g Two semigroups having the same generating set G and if and only if f, g have the same number of image ele- the same relations are isomorphic, since the mapping ments, and f ᏾ g if and only if their images are the same. f (x1 ◦ x2 ◦ · · · ◦ xn) = x1 ∗ x2 ∗ · · · ∗ xn will be a homo- In a finite semigroup, ᏶ = Ᏸ. The entire semigroup is morphism where ◦, ∗ denote the two operations. The iden- always broken into ᏶-classes, which are partially ordered tity of the sets of relations guarantees that this is well de- by divisibility. The Ᏸ-classes, which are in these, are bro- fined, that is, if x1 ◦ x2 ◦ · · · ◦ xm = y1 ◦ y2 ◦ · · · ◦ yn, then ken into Ᏼ-classes Hi j = Ri ∩ L j , where Ri , Li are the f (x1 ◦ x2 ◦ · · · ◦ xm) = f (y1 ◦ y2 ◦ · · · ◦ yn). ᏾, ᏸ-classes in a given Ᏸ-class. For any set of generators G and set R of relations, a The Ᏼ-classes can be laid out in a matrix (Hi j ) called semigroup S can be produced having these generators the eggbox picture of a semigroup. There exists a 1–1 and satisfying the relations such that if f (G) → T is any correspondence between any two Ᏼ-classes. function such that for every relation x1 ◦ x2 ◦ · · · ◦ xm = A left ideal in a semigroup S is a set Ᏽ ⊂ S such that y1 ◦ y2 ◦ · · · ◦ yn the relation f (x1) ∗ f (x2) ∗ · · · ∗ f (xm) for all x ∈ S, y ∈ Ᏽ the element xy ∈ℑ. Right and two- = f (y1) ∗ f (y2) ∗ · · · ∗ f (yn) holds then f extends to a sided ideals are similarly defined by yx ∈ℑ, yxz ∈ℑ homomorphism S → T . for all y, z ∈ S, respectively. An element x generates the To produce S we take the set of all “words” x1x2 · · · xm principal left, right two-sided ideals Sx = {yx : y ∈ S}, in G, that is, sequences of elements in G. Two “words” are x S = {xy : y ∈ S}, Sx S = {yxz : y, z ∈ S}. Two elements multiplied by writing one after the other x1x2 · · · xm y1y2 are ᏸ-, ᏾-, ᏶-equivalent (assuming S has an identity, oth- · · · yn. We define an equivalence relation on words by erwise add one) if and only if they generate the same left, w1 ∼ w2 if w2 can be obtained from w1 by a series of re- right, or two-sided ideals, respectively. placments of a x1x2 · · · xnb by ay1y2 · · · ymb, where a, b are words (or are empty) and x1x2 · · · xn = y1y2 · · · ym is a relation in R. Then S is the set of equivalence classes of C. Binary Relations and Boolean Matrices words. The semigroup S is called the semigroup with genera- The set of binary relations on an n-element set under com- tors S and defining relations R. The fact that multiplication position forms a semigroup that is isomorphic to Bn, the is well defined in S follows from the fact that a ∼ b is a semigroup of n-square Boolean matrices under Boolean congruence. This means that if a ∼ b then for all words matrix multiplication. A 1 × n(n × 1) Boolean matrix is x, ax ∼ bx and xa ∼ xb, and that ∼ is an equivalence called a row (column) vector. Two vectors are added or relation. multiplied by a constant (0 or 1) as matrices: (1, 0, 1) + For all semigroup homomorphisms f : S → T the re- (0, 1, 0) = (1, 1, 1), 0v = 0, and 1v = v. A set of Boolean lation f (x) = f (y) is a congruence. Conversely any con- vectors is called a subspace if it is closed under sums gruence gives rise to a homomorphism from S to the set and contains 0. The span of a set W of Boolean vectors of equivalent classes. is the set of all finite sums of elements of W , including 0. P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 Algebra, Abstract 441 The row (column) space of a Boolean matrix is the space If an Ᏼ-class contains an idempotent, then it will be spanned by its row (column) vectors. Two Boolean ma- closed under multiplication, and multiplication by any of trices are ᏶- (also Ᏸ-) equivalent if and only if their row its elements gives a 1–1 onto mapping from it to itself. spaces are isomorphic. The basis for a space of Boolean Therefore, it forms a group. Conversely any group in a vectors spanned by S is {x ∈ S : x ≠ 0 and x is not in the semigroup lies in a single Ᏼ-class since under multiplica- span of S\{x}}. Two subspaces are identical if and only if tion any two elements are ᏸ-equivalent and ᏾-equivalent. their bases are the same. Bases for the row, column spaces of a Boolean matrix are called row, column bases. Two E. Finite State Machines Boolean matrices are ᏸ- (᏾-) equivalent if and only if their row (column) bases (and so spaces) coincide. The theory of automata deals with different classes of n A semigroup of the form {x } is called cyclic. Every theoretical machines representing robots, calculators, and cyclic semigroup is determined by the index k = inf{k ∈ similar devices. The simplest are the finite state machines. + k+m k + Z : x = x for some m ∈ Z }, and the period d = There are two essentially equivalent varieties: Mealy ma- + k+d k inf{d ∈Z : x = x }. The set of powers k, k + 1, . . . re- chines and Moore machines. peat with period d. If an n-square Boolean matrix A is re- A Mealy machine is a 5-tuple (S, X, Z, ν, µ), where 2 3 n−1 flexive (A ≥ I ), then A = AI ≤ A ≤ A ≤ · · · ≤ A = S, X, Z are sets, ν a function S × X to S, and µ a function n A , an idempotent. Here I denotes the identity matrix. S × X to Z. The same definition holds for Moore machines Also if A is fully indecomposable, meaning that there ex- except that µ is a function S to Z. ists no nonempty K, L ⊂ {1, 2, . . . , n} with |K | + |L| = n, Here S is the set of internal states of the machine. For where |S| denotes the cardinality of a set S, and ai j = 0 for a computer this could include all posibilities as to which n−1 i ∈ K, j ∈ L, then A = J and so A has period 1 and in- circuit elements are on or off. The set X is the set of inputs, dex at most n − 1. Here J is the Boolean matrix all entries which could include a program and data. The set Z is the of which are 1. set of outputs, that is, the desired response. The function In general, an n-square Boolean matrix has period equal µ gives the particular output from a given internal state to the period of some permutation on {1, 2, . . . , n} and and input. The function ν gives the next internal state 2 index at most (n − 1) + 1. resulting from a given state and input. For a computer this is determined by the circuitry. For example, a flip-flop will change its internal state if it receives an input of 1; D. Regularity and Inverses otherwise the internal state will remain unchanged. An element x of a semigroup S is said to be a group inverse A Mealy machine to add two n-digit binary numbers, of y if S has a two-sided identity and xy = yx = . A a, b can be constructed as follows. Let the i th digits of semigroup in which every element has a group inverse is a i , bi be inputs, so X = {0, 1} × {0, 1}. Let the carry from called a group. In most semigroups, few elements are in- the i − 1 digit be c i . It is an internal state, so S = {0, 1}. vertible in this sense. A function is invertible if and only if The output is the i th digit of the answer, so Z = {0, 1}. it is 1–1 onto. A Boolean matrix in invertible if and only The function ν gives the next carry. It is 1 if and only if if it is the matrix of a permutation (permutation matrix). a i + bi + ci > 1. The function µ gives the output. It is 1 if There are many weaker ideas of inverse. The two most and only if a i + bi + ci is odd. Figure 5 gives the values important are regularity yxy = y and Thierrin–Vagner in- of ν and µ. verse yxy = y and xyx = x. An element y has a Thierrin– With a finite state machine is associated a semigroup Vagner inverse if and only if it is regular: If yxy = y of transformations, the transformations f x (s) = ν(s, x) of then xyx is a Thierrin–Vagner inverse. A semigroup in which all elements are regular is called a regular semigroup. The semigroups of partial transformation, transformations, and n-square matrices over a field are regular. In the semigroup of n-square Boolean matrices, an element is regular if and only if its row space forms a distributive lattice as a poset. An idempotent is an element x such that xx = x. An element x is regular if and only if its ᏸ-equivalence class contains an idempotent: if xyx = x then xy is idempotent. The same holds for ᏾-equivalence classes. Therefore, if two elements are ᏸ or ᏾-equivalent (therfore Ᏸ-equivalent), one is regular if and only if the other is. FIGURE 5 Machine to add two binary numbers. P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 442 Algebra, Abstract the state space, and all compositions of them fx 1x2···xn (s) = symbol. If (x, y) ∈ ρ, we are allowed to change any oc- fx 1 ( fx2 · · · fxn (s)). This is called the semigroup of the currence of x with y. Members of W are called terminals. machine. Consider logical formulas involving operations ∨, ∼ Two machines are said to have the same behavior if and variables p, q, r . Let W = {p, q, r, ∨, (,), ∼}, N = there exists a binary relation R from the initial states of one {ψ}. We derive formulas by successive substitution, as machine to the intial state of the other such that if (s, t) ∈ S ψ, (ψ ∨ψ), ((ψ ∨ψ)∨ψ), ((ψ ∨ψ)∨ ∼ ψ), ((p∨ψ)∨ ∼ then for any sequence x1, x2, . . . , xk of inputs machine 1 ψ), ((p ∨ q) ∨ ∼ ψ), ((p ∨ q) ∨ ∼ r). ∗ in state s gives the same sequence of outputs as machine An element y ∈ (N ∪ W) is said to be directly de- ∗ 2. A machine M1 equivalent to a given machine M having rived from x ∈ (N ∪ W) if x = azb, y = awb for some ∗ a minimal number of states can be constructed as follows. (z, w) ∈ ρ, a, b ∈ (N ∪ W) . An indirect derivation is a Call two states of M equivalent if, for any sequence of sequence of direct derivations. Here ρ = {(ψ, p), (ψ, q), inputs, the same sequence of outputs is obtained. This (ψ, r), (ψ, ∼ψ), (ψ, (ψ ∨ ψ))}. gives an equivalence relation on states. Then let the states The language determined by a phrase structure grammar ∗ of M1 be the equivalence classes of states of M. is the set of all a ∈ W that can be derived from ψ. No finite state machine M can multiply, as the above A grammar is called context free if and only if for all machine adds binary number of arbitrary length. Suppose (a, b) ∈ ρ, a ∈ N, b ≠ e0. This means that what items can such a machine has n states. Then suppose we multiply be substituted for a given grammatical element do not de- k the number 2 by itself in binary notation (adding zeros pend on other grammatical elements. The grammar above in front until the correct number of digits in the answer is is context free. achieved). Let fx be the transformation of the state space A grammar is called regular if for all (a, b) ∈ ρ we have given by inputs of 0, 0 and let a be the state after the inputs a ∈ N, b = tn, where t ∈ W, n ∈ N, or n = e0. This means i 1, 1. The inputs 0, 0 applied to state f x (a) give output 0 at each derivation we go from t1t2 · · · tr n to t1t2 · · · tr tr+1m, for i = 0, 1, 2, . . . , k − 2 but output 1 for i = k − 1. Yet the where ti are terminals, n, m nonterminals, (n, tr+1m) ∈ ρ. transformation f on a set of n elements will have index at So we fill in one terminal at each step, going from left to k−1 most n. Therefore, if k > n then f (a) will coincide with right. The grammar mentioned above is not regular. x j fx (a) for some j < k − 1. This is a contradiction since one To recognize a grammar is to be able to tell whether or ∗ yields 0 output, the other 1 output. It follows that no such not a sequence from W is in the language. A grammar is machine can exist. regular if and only if some finite state machine recognizes it. The elements of W are input 1 at a time and outputs are “yes, no,” meaning all symbols up to the present are or F. Mathematical Linguistics are not in the language. Let the internal states of the ma- We start with a set W of basic units considered words. chine be in 1–1 correspondence with all subsets of N, and Mathematical linguistics is concerned with the formal let the initial state be ψ. For a set S1 of nonterminals and theory of sentences, that is, sequences of words that input x let the next state be the set S2 of all nonterminals z are grammatically allowed and the grammatical structure such that for some u ∈ S1, (u, xz) is a production. Then at of sentences (or longer units). This is syntax. Meaning any time the state consists of all nonterminals that could (semantics) is usually not dealt with. occur after the given seqence of inputs. Let the output be ∗ For a set X, let X be the set of finite sequences from “yes” if and only if for some u ∈ N, (u, x) ∈ ρ. This is if X including the empty sequence e. For instance, if X is and only if the previous inputs together with the current ∗ {0, 1}, then X is {e, 0, 1, 00, 01, 10, 11, 000 · · ·}. For a input form a word in the language. more important example, we can consider the family of For the converse, if a finite state machine can recognize all sequences of logical variables p, q, r , and ∨ (or), ∧ a language, let W be as in the language, N be the set of (and), (,) (parentheses), → (if then), ∼ (not). The set of internal states, ψ the initial state, the productions the set logical formulas will be a subset of this. of pairs (n1, xn2) such that if the machine is in state n1 A phrase structure grammar is a quadruple (N, W, and x is input, state n2 is the next state, and the set of ρ, ψ), where W (set of words) is nonempty and finite, pairs (n1, x) such that in state n1 after input x the machine N (nonterminals) is a finite set disjoint from W, ψ ∈ N, answers “yes.” ∗ ∗ ∗ and ρ is a finite subset of ((N ∪ W) \W ) × (N ∪ W) . A further characterization of regular language is the ∗ The set N is a set of valid grammatical forms involving Myhill–Nerode theorem. Let W be considered a semi- ∗ abstract concepts such as ψ (sentence) or subject, predi- group of words. Then a language L ⊂ W is regular if ∗ ∗ cate, object. The set ρ (productions) is a set of ways we can and only if the congruence {(x, y) ∈ W × W : axb ∈ L substitute into a valid grammatical form to obtain another, if and only if ayb ∈ L for all a, b ∈ L} has finitely many more specific one. The element ψ is called the starting equivalence classes. This is if and only if there exists a P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 Algebra, Abstract 443 ∗ finite semigroup H, a homomorphism h : W → H is onto, −1 and a subset S ⊂ H such that L = h (S). III. GROUPS A. Examples of Groups A group is a set G together with a binary operation here denoted ◦ such that (1) ◦ is a function G × G → G (closure); (2) ◦ is associative, for example, (x ◦ y) ◦ z = x ◦ (y ◦ z); (3) there exists a two-sided identity such that FIGURE 6 Multiplication table of the dihedral group. ◦ x = x ◦ = x for all x ∈ G; (4) for all x ∈ G there exists y ∈ G with x ◦ y = y ◦ x = . Here y is known as an inverse of x. From now on we suppress ◦. n. It is isomorphic to the group of functions Zn → Zn of For any semigroup S with a two-sided identity , let the form f (x) = ±x + k. Its multiplication table is given S∗ = {x ∈ S : xy = yx = for some y ∈ S}, the set of in- in Fig. 6, where xi is x + i − 1 if i > n + 1, −x + i − n − 1 vertible elements. The element y = x−1 is unique since if if i > n. ∗ y1x = , xy2 = then y1 = y1xy2 = y2. Then S satisfies −1 −1 −1 −1 closure since (ab) = b a and is a group where a B. Fundamental Homomorphism Theorems is an inverse of a. A group with a finite number of elements is called a A group homomorphism is a function f : G → H satis- finite group; otherwise, it is called an infinite group. If a fying f (xy) = f (x) f (y) for all x, y in G. If f is 1–1 group G is finite, and G contains n elements, we say that and onto it is called an isomorphism. For any group G the order of G is n and we write |G| = n. If G is infinite, of order n there is a 1–1 homomorphism G → ᏿n; la- we write |G| = ∞. In general, |G| denotes the cardinality bel the elements of G as x1, x2, . . . , xn. For any x in G, of a set G. the mapping a → xa gives a 1–1 onto function G → G, The group of all permutations of {1, 2, . . . , n} is called so there exists a permutation π with xxi = xπ(i). This is ᏿n, the symmetric group of degree n. The degree is the a homomorphism since if x, y are sent to π, φ, we have number of elements in the domain (and range) of a permu- yxxi = yxπ(i) = xφ(π(i)). There is an isomorphism from ᏿n tation. Therefore, ᏿n has degree n, order n!. A subgroup to the set of n-square permutation matrices (either over R of ᏿n is formed by the set of all transformation of the form or a Boolean algebra), and it follows that any finite group 1→k +1, 2→k +2, . . . , n − k → n, n − k + 1 → 1, . . . , can be represented as a group of permutations or of in- n → k. vetible matrices. For any set P of subsets of n-dimensional Euclidean Groups can be simplified by homomorphisms in many n space E , let T be the union of the subsets of P. A sym- cases. The determinant represents the group of nonsin- metry of P is a mapping f : T → T such that (S–1) for gular n-square matrices in the multiplicative group of x, y ∈ T , d(x, y) = d( f (x), f (y)), where d(x, y) denotes nonzero real numbers (it is not 1–1). For a bounded func- ∫ the distance from x to y; and (S–2) for A ⊂ T , A ∈ P if tion g, g f dx is a homomorphism from the additive and only if f (A) ∈ P. That is, f preserves distances and group of integrable functions f to the additive R. sends the subsets C of P, for example, points and lines, so The kernel of a homomorphism f : G → H is other subsets of P. The inverse off is its inverse function, {x : f (x) = }. Every homomorphism sends the identity which also satisfies (S–1) and (S–2). to the identity and inverses to inverses. Existence of in- The sets R, Z, Zm under addition are all groups. verses means we can cancel on either side in groups: −1 −1 The group Zm, the group of permutations {1 → k + 1, If ax = ay then a ax = a ay, x = y, x = y. There- 2 → k + 2, . . . , n → k}, and the group of rotational sym- fore, the identity is the unique elements satisfying = . metries of an n-sided regular polygon are isomorphic. That Under a homomorphism f ( ) = f ( ) = f ( ) f ( ) so is, there exists a 1–1 correspondence f between any two f ( ) is an identity. So is in the kernel. A ho- such that f (xy) = f (x) f (y) for all x, y in the domain. A momorphism is 1–1 if and only if is its only el- −1 −1 group isomorphic to any of these is called a cyclic group ement: If f (x) = f (y), then f (xy ) = f (x) f (y ) = −1 −1 of order m. f (x) f (y) = f (x) f (x) = . The group of all symmetries of a regular n-sided poly- If x, y are in the kernel, so is xy since f (xy) = −1 gon has order 2n and is called the dihedral group of order f (x) f (y) = = . If x is in the kernel, so is x . Any P1: FVZ Revised Pages Encyclopedia of Physical Science and Technology EN001C.19 May 7, 2001 13:42 444 Algebra, Abstract subset of group that is closed under products and inverses For any homomorphism f : G → H with kernel K , im- is a subgroup. Both the kernel and image of a homomor- age M every equivalence class of K is sent to a single phism are subgroups. element. That is, f gives a mapping G/K → M. This −1 The mapping cg : G → G defined by cg(x) = gxg is mapping is an isomorphism. called a conjugation. An isomorphism from a group (or If A, B are normal in a group G, and B is a subgroup other structure) to itself is called an automorphism. Since of A, then −1 −1 −1 cg(xy)=gxyg = gxg gyg , cg is a homomorphism. G/B If h = g−1, then cg is the inverse of ch. Therefore, the map- = G/A A/B pings cg are automorphisms of a group. They are called If A is a normal subgroup and B is any subgroup of G, the inner automorphisms. This gives a homomorphism and AB = {ab : a ∈ A, b ∈ B} then, of any group to its automorphisms, which also form a group, the automorphism group. A subgroup N is normal AB B = if cg(x) ∈ N for all x ∈ N and g ∈ G. The kernel is always A A ∩ B a normal subgroup. If f : G → H has kernel K and is onto, the group G is C. Cyclic Groups said to be an extension of K by H. Whenever there exists k a homomorphism f from K to Aut(H) for any groups Let G be a group. If there exists a ∈ G such that G = {a : K, H, a particular extension exists called the semidirect k ∈ Z}, then G is called a cyclic group generated by a, and product. Here Aut(H) denotes the automorphism group a is called a generator of G. For addition, the definition of H. This has as its set K × H and products are defined becomes: by (k1, h1)(k2, h2) = [k1k2, f (k2)(h1)(h2)], k1, k2 ∈ K and There exists a ∈ G such that for each g ∈ G there exists h1, h2 ∈ H. The dihedral group is a semidirect product of k ∈ Z such that g = ka. Z2 and Zm. A group G is said to be commutative when xy = yx The groups K, H will generally be simpler than G. A for all x, y in G. Commutative groups are also called −1 theory exists classifying extensions. If a group has no nor- Abelian. In a commutative group, gxg = x for all mal subgroups except itself and { }, it is called simple. g ∈ G. Therefore, every subgroup of a commutative n The alternating group, the group of permutations in ᏿n group is normal. Let R denote the set of all n-tuples of n whose matrices have positive determinant, is simple for real numbers. Then (R , +) is a commutative group. For n > 4. Its order is n!/2. For n odd, the group of n-square any groups G1, G2, . . . , Gn, the set G1 × G2 × · · · × Gn real-valued invertible matrices of determinant 1 is sim- under componentwise multiplication (x1, . . . , xn) × ple. For n even a homomorphic image of ᏿n with kernel (y1, . . . , yn) = (x1y1, . . . , xn yn) is a group called the {I, −I } is simple, where I is an identity matrix. All finite Cartesian or direct product of G1, G2, . . . , Gn. If all Gi and finite-dimensional differentiable, connected simple are commutative, so is the direct product. For any indexed groups are known and most are constructed from groups of family Gα of groups, coordinatewise multiplication matrices. makes the Cartesian product a group ×αGα. The subset For any subgroup H of a group G there exist equiva- of elements (xα) such that {α : xα = } is finite is also a −1 −1 lence relations defined by x ∼ y if xy ∈ H(y x ∈ H). group sometimes called the direct sum. These equivalence classes are called left (right) cosets. A set G0 ⊂ G is said to generate the set −1 The left (right) coset of a is Ha = {ha : h ∈ H} {x1x2 · · · xn : xi ∈ G0 or x i ∈ G0 for all i and n ∈ Z}. (aH = {ah : h ∈ H}). There exists a 1–1 correspondence Every finitely generated Abelian group is isomorphic to H → Ha given by x → xa (x → ax for right cosets). a direct product of cyclic (1 generator) groups. Therefore, all cosets have the same cardinality as H. If A group of words on any set X is defined as the a(1) a(2) a(n) [G : H] denotes the number of right (or left) cosets, then set of finite sequences (with ) x 1 x2 · · · xn , xi ∈ X, |G| = |H|[G : H], which is known as Lagrange’s theorem. a(n) ∈ Z. We can reduce such words until a(i) ≠ 0, So the order of a subgroup of a finite group must divide xi ≠ xi+1 by adding exponents if and only if they have the the order of the group. If G is a group of prime order p, same reduced form. Equivalence classes of words form a n let x ∈ G, x ≠ . Then {x } forms a subgroup whose order group F. n divides and must therefore equal p. So G = {x }. Relations among generators in a group can be writ- a(1) a(2) a(n) If N is a normal subgroup and x ∼ y, for all g ∈ G then ten in the form x 1 x2 · · · xn = . For any set G0 and −1 −1 −1 gx ∼ gy and xg ∼ yg since gxy g and xy are in set of relations R in the elements of G0, there exists a −1 N if xy in N. It follows that if a ∼ b and c ∼ d, then group G defined by these relations such that any map- ac ∼ bd. Therefore, the equivalence classes form a group ping f : G0 → H extends to a unique homomorphism G/N called the quotient group. Its order is |G|/|N|. g : G → H if the relations hold in H. The group G is