Modern Algebra MATH 6302
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Date Created: 09/19/15
Modern Algebra Math 63026303 Klaus H Kaiser Chapter 1 Algebraic Systems 11 Operations Algebraic Systems An operation f on the set A is a map f z A A A Where n is a natural number n 2 0 The number n is called the arity of n szgtA andfisjustamaponA7 n 2 f z A2 A A and fis a binary operation on Al The case n 0 deserves special attention For any set S one has that AS ala S A A In particular7 for n 0717 i i 7 n 71 7 A alaz07l7ui7nilHAaa07iu7an1layeA and this is the set of all n 7 tuples a07i i i 7 an11 of elements in A An niary operation assigns to any nituple a07 i i i 7 an11 of elements in A an element fa07 i i i 7 an11 as operation value N0W7 A0 ala 0 A A A nullary operation assigns to the empty map an element a E A i A nullary operation is therefore called a constant For A 0 one has that A0 and A D for n gt 0 That is7 only operations of positive arity existi An algebraic system consists of 1 a set A7 2 a family of n 7 ary operations on Al We use the notation A A7 faker and A 710161quot is called the similarity type of Al Examples 1 Z Z7f17f27f3 Where f11Z2 A Z7 n7m gt gt nm7 f2 Z A Z7 1 gt gt 717 f3 A Z7 gt gt 0 The binary operation fl is the addition7 the unary operation f2 is the additive inverse and f3 is the zero We use for binary operations most of the time the symmetric notation zfy instead of fz7 2 Let S be any set MapS is the algebra of all maps of S into itself with composition 0 as a binary operation and ids as a nullary operation Thus we get the algebraic structure Maids Me I 5 A S 07 ids 001571 WW 3 Similarly as before we de ne Bij590l9015 E s o ids The familiar algebras of integers real numbers and complex numbers Z Z70 71 R R n oy 3971 C 07770 3971 are algebras of the same type A 2 l 0 2 0 They are similar Generalizations of the concept of an algebraic system are partial algebras The operations are not everywhere de nedl On the set of real numbers 1 R0 7gt R z gt gt 11 can be added as a partial unary operationl in nitary algebras These are algebras with operations of in nitary arityl Taking the limit of an in nite sequence may be considered as a partial in nitary operation the operation is de ned only for convergent sequences The projections pie A 7gt A aim61 gt gt aio are in nitary operations if I is in nite multivalued algebras The operation values are subsets of A ie f z A 7gt 73A fa1l l l an Q 73A One has that cardfa1 l l i an 1 iff f is an operation and cardfa1 l l l an S 1 iff f is a partial operationl relational algebras A relational system is a set A together with a family f teT of n 7 ary operations and a family R5565 of ms 7 ary relations R5 where one has that each R5 Q Ami An n 7 ary operation f corresponds uniquely to an n l 7 ary relation Rf Rf a1l l l anan1lan1 fa1 l l i an graphf An Example of a relational systems is Z ZHn oy 3971 S Any algebraic system may be considered as a relational system where S 0 Also nitary partial algebras are relational systemsl Note that the operations of the algebraic system as well as the relations of a relational system are nitary but that the number of operations of an algebra A is in general in nite Example Let V be a vector space over the eld of real numbers Then multiplication of a vector v by a real number a may be perceived as a unary operation and we have for each a E R the unary operation 1 gt gt alvl As an algebraic system such a vector space looks like V 7 70 10461 12 Homomorphisms of Algebras Subalgebras and Direct Prod ucts Let A and B be algebras of the same type A A A7 ftteT7 B B7 926T Where the arity of ft is equal to the arity of g t 6 T A map go A A B of the underlying set A of A to the underlying set E of B is called a homomorphism if for every t E T one has that fzao77777an71 9280a077 77780017771 This means for a binary operation eigi multiplication Which is denoted in both algebras by 77 z ao a1 ao WW1 For a unary operation eg a multiplicative inverse 1 this reads as ail a 1 For a nullary operation say at a and b b this is a b Let for a homomorphism go A A B Ev be the equivalence that is induced by go that is al N a m0dE4p and only al a An equivalence relation E on an algebra A is a congruence relation if al N a3 a2 N a2Hi ant N agt yields fta1uiam N fta1i agt Proposition 11 Let go A A B be a homomorphism of algebras Then the equivalence for go Ego a1a2lgoa1 a2 is a congruence relation on PROOFi Assume al N a3 a N a 2i ant N agti This is al a liugoam agt Therefore 900301177 7 7 am 9490011 7 7 7 70077 97soa 177 7 7 790012 90030137 7 7 7 77 This is fta1uiam tha1Hiati D Proposition 12 Let E be a congruence relation on the algebra A Then there is exactly one algebraic structure on the set AE of equivalence classes for E such that the canonical projection qE A A AE becomes a homomorphism PROOFi We rst show uniquenessi lf qE a A a ME is homomorphic then M017 7 7 7 70m ltla1l77 7 7 7 lamb 130112011 7 774Eam 4Efza177777am7 Therefore if C1 a1i i Cm ant then necessarily filt0177 7 7 7 Cm lfzlta177 7 7 7an l Assume now for the congruence E on A that C1iH Cm a1iu amD ftgah i ant then because E is a congruence the choice of a1 6 C1 i i a Cm does not matter ft is properly m E 7 de ned by means of representatives The map qE A A AE is homomorphic 4Efza177777am fm4Ea1777774Eam is true by the de nition of D Proposition 13 Let go A A B be a homomorphism between algebras with associated congruence Ev a N a a a Then there is a unique injective homomorphism AE A B such that o qE so A E AE L B A L B PROOFl We only have to show that is homomorphicl But lt lt0170mgtgt amp1m ngt ltqufza17wan 9003011 a gltsolta1gt w gm 0 qEWgtlta1gtmlt 0 mm gtlt lt01gt are a A bijective homomorphism is called an isomorphisml Let go A A B be an isomorphism between algebras Then soil B A A is homomorphic so lyzblwbm fzso 1b17wsf1bmiff fzsf1bl7 04077 92517Awbniff 92soso 1b1 79090 1bn 92517Awbn Corollary 14 Homomorphism Theorem Let go A A B be a surjective homomorphism between algebras Then B is isomorphic to a factor algebra of A Note7 that the composition of homomorphisms is a homomorphism and that the identity on an algebra is a homomorphisml Thus7 the class of algebras of similarity type A is a category With the algebras as objects dots and the homomorphisms as morphisms arrows lt is very easy to see that for every algebra A7 the intersection of congruence relations is a congru ence Recall that an ordered set P7 S is a complete lattice if every subset S of P has a largest lower bound or a smallest upper bound Thus we have Proposition 15 For every algebra A the congruence relations on A form a complete lattice ConAl A aala E A is the smallest congruence on A and AA 2 Al A X A is the largest congruence on A and one has that AA X A 1A iel7 the oneelement algebra of type A Let A be an algebra and let C be a subset of Al C is said to be closed if a17 an E C implies that fza1 am 6 C ln particular7 a closed subset contains all constants Proposition 16 Let C be a closed subset of the algebra A Then there is exactly one algebraic structure g on C such that the inclusion icAZClt gtA is homomorphic PROOF We rst prove uniqueness iyza17m7am fzia17iam fta17an 92a17m7am That is 9 leCm On the other hand for any closed subset C of A we may de ne gta1 am fta1 am a1ant E C and i C lt gt A is homomorphic B Any injective homomorphism is called an embedding A closed subset becomes an algebra such that the inclusion is an embedding Any closed subset C with the operations ft restricted to C is a subalgebra It is easy to see that for any algebra A the intersection of closed subsets is closed Proposition 17 For every algebra A the subalgebras ofA form a complete lattice SubA A is the largest element of SubA and the smallest element of A is Z if there are no constants otherwise it is the subalgebra C0 that is generated by all constants C0 C m C where M0 CLO ftolto E T and mo 0 CESubA 203 e a Let M0 be any subset of the algebra A Then de ne CAM0 0 CESubA CQMO Then CA M0 is the smallest subalgebra of A that contains M0 For any set M and ntary operation ft of A de ne f M ala E A a fta1 am a1 am 6 M M07 M1 U HMS UM07 M2 U ftltMiLtUM1 yields tET tET M0 M1 M2 Then let M U My We notice 0 M is closed Let a1 7am E M Then there is some k such that al am 6 Mk This implies that fta1 am 6 Mk1 E M o M contains M0 0 Let C be any subalgebra containing M0 Then C contains M We have C 2 M0 and assume that C 2 Mk Let a E M1644 Then a E Mk or one has that a fta1ant with a1 am 6 Mk In both cases we get that a E C Hence7 M Proposition 18 C CA de nes an algebraic closure for subsets M of the algebra A That is o C is extensive ie CM 2 M o C is monotone ie If M1 Q M2 then CM1g CMg o C is idempotent ie CCM Moreover a E CM if and only if there is a nite subset F of M such that a E CF CM U FgM CF F rm PROOF It is quite obvious that C satis es the properties of a closure operator We only have to show the algebraicity of C We need to show that U Cm is closed Let a17 am 6 U CF We then have that 315 a1 6 CF17am E CFm implies that a17 am 6 CF1 U H U Fm CF where7 of course7 F is nite But then fzltahwamgtecltFgtg U W FgM F finite Now7 U CF is a closed subset that contains M This is clear because for any element a E M one has that a E Ca Q U CF and therefore M Q U CF and therefore CM Q U CF The converse inclusion is of course obvious D Proposition 19 Let Alig Ai7 fitteTl61 be a family of similar algebraic systems Let AHAZ a a1gt LJAZ7 ai 6 Ai i6 i6 be the cartesian product of the carrier sets Ai of the algebras Ai Then there is exactly one algebraic structure f teT on the set A such that all projections pi A A i a gt gt ozi7 are homomorphic PROOF We rst have to show uniqueness Let 11 7am E A and let a 1610117 701m Then one has that Mi Ma 10103011 A A A 7am fl0ial7 A A A 7piam filta1i7ami and that is I a Mime fla1i A Avan iiei This yields7 I fza17wam fla1i A wan ltigtgtgti61 On the other hand7 if we de ne ft on the cartesian product A by this last formula7 then 1010301 7am 13110110 v 7ami flPia17m7piam shows that the projections pi are homomorphismsi D Note that we have in particular ct cal61 for nullary operations C For any algebra A7 and any set S A5 is called the direct power of Al The plane R2 is a typical example of the second power of the vector space R Proposition 110 Let goi B A Ai i E I be an initial family of homomorphisms Then there is exactly one homomorphism go B A A H Ai such that pi a go 301 i E It i6 PROOFi One de nes so 11 I B A A 901 soibier D i6 De nition 1 Gt Birkhoff A class 73 of algebras is called primitive if it is closed 0 under the formation of direct products 0 under taking homomorphic images 0 under taking subalgebrasi Groups and rings are examples of primitive classesi However7 elds are not closed under direct productsi De nition 2 At It Mal cev A class Q of algebras is quasiprimitive if it is closed under 0 isomorphic copies 0 direct productsi Cancellation semigroups are an example of a quasiprimitive class that is not primitivei Remarks on Ordered Sets and Lattices A relational system 07 S is called an ordered set if the relation S is refleacive7 ii transitive and if iii antisymmetry holdsi That is i zSz holds for allze 0 ii lfzSyandySzthenzSzi iii lfzSyandySzthenzyi For I f y but I S y one writes I lt y Also7 I S y means the same as y 2 I An ordered set is totally ordered if one also has trichotomy iv Eitherxgyorygxorxyi Prominent examples for ordered sets are the natural numbers with divisibility N l and the power set 735 Q for the set 5 The set of real numbers with their ordinary ordering is the prototype of a totally ordered set Very often ordered sets are called partially ordered and totally ordered sets are called linearly ordered A relation that is re exive and transitive is called a quasiordering The integers with division are an example nlm and mln only yields n i If S is a subset of the ordered set 0 then u is called an upper bound for S if u 2 s holds for all s 6 5 An upper bound that actually belongs to S is called the maximum of 5 A maximum if it exists is of course by antisymmetry unique If u is an upper bound for S and if v 2 u then v is an upper bound for 5 That is the upper bounds for a subset S for the ordered set 0 form an upper end of 0 Lower bounds and minima are de ned similarlyi By 77default every element of O is an upper as well a lower bound of the empty set If the set of upper bounds for S has a minimum then this minimum is called the supremum of S or the least upper bound of S supS minulu is an upper bound for 5 Similarly the of S or largest lower bound is the maximum of all lower bonds for 5 An ordered set is called a complete lattice if every subset S has a supremum as well an in mumi By the very de nition the in mum of the empty set must be if it exists the maximum of 0 Similarly the supremum of the empty set must be if it exists the minimum of 0 Only for the empty set the in mum may be greater than the supremumi An ordered set is bounded if it has a maximum as well a minimumi The following proposition is an easy but useful fact The proof is an easy exerciser Proposition 111 Assume that every subset S of the ordered set 0 has an Then every subset S of O has a supremum An ordered set 0 is called a complete lattice if every subset S has an in mum and then as well a supremumi The power set 735 of a set S is the prototype of a complete latticei Here the in mum of a subset iieia collection C of subsets of S is the intersection of C and the supremum is the union of C A collection C of subsets of S is called a closure system of S if the intersection of every subcollection S of C belongs to Cl As a corollary to the last proposition we state Proposition 112 Let T be a closure system of subsets of the set S Then T Q is a complete lattice An ordered set is a lattice if every nite nonempty subset has an in mum as well a supremumi The natural numbers with divisibility form an example of a complete lattice The in mum of a set of natural numbers is the greatest common divisor and the supremum is the lowest common multiple The number 0 is the maximum of N l and l is the minimum of N An ordered set is called well ordered if every subset has a minimumi The natural numbers are well ordered by g It is an axiom of set theory that every set can be well ordered This statement is equivalent to the Axiom of Choice The axiom of choice is also equivalent to Zorn s Lemma Assume that every totally ordered subset C of the ordered set 0 S has an upper bound in 0 Then 0 has a maximal element An element m of O is maximal if s 2 m implies that s mi Finally we remark that if R is an ordering the dual relation aR b iff bRa is also an ordering Instead of saying that a certain map go A A B is an order reversing isomorphism from A to B we say that it is an order isomorphism from A to B3 Note that go is an order isomorphism if it is bijective and go as well as go 1 are order preserving Chapter 2 Basic Facts about groups rings and elds Modules and vector spaces 21 Groups A groupoid is an algebraic system With just one binary operation A A groupoid becomes a semigmup if the operation is associative ie one has for all z y 2 E A IyzIy2 If A is a groupoid then 6 E A is called a unit for 77 if one has for all z E A A groupoid can have at most one unit Assume that e and e are units We then have 6 e 6 because 6 is a leftunit and e e 6 because 6 is a ghtunit Hence 6 equot A semigroup With unit is called a monoid S e Where we consider the unit 6 as a nullary operation Let S e be a monoidi Then I E S is called invertible if there is an 1 E S such that An element 1 of a monoid can have at most one inversei Assume that z is a leftinverse and that z is a ghtinverse of 1 ie 1 z e and z z er But then 1zez11 Izz ez z The inverse of an element 1 in a monoid if it exists is denoted as I ll A group is a monoid Where every element has an inverse Therefore the algebraic system G G 1e is a group if 0 the binary operation is associative 1y2Iyz o The constant e is a unit for 1 o The unary operation associates for every I E G the inverse o A group is abeliah if the binary operation is commutative z y y z A commutative group is also called a module The binary operation then is denoted as and called addition The unit is called zero and 7 is the operation that takes the additive inverse Examples 1 Z Z 70 is a module 2 R R 0 1 1 is a group ie the multiplicative group of the reals 3 N N 0 and N N l are monoids Theorem 21 The class of all semigroups is a primitive class of algebras The same is true for the class of all mohoids the class of all groups and the class of abeliah groups PROOF Let C be a closed subset of the semigroup S Then C C is obviously a semigroup Let go C A A be a surjective homomorphism from a semigroup to a groupoid Let abc E A Then a z b y c z and a 39 b 39 C 801 39 809 39802 901 y 902 8019 392 80139 92 I 9092 90190y902 a b e Let Ai i61 be a system of groupoids All operations are denoted by Let 15 6 H 14139 Recall i6 that a I A U Ai ai 6 14139 We also Write a aiie and consider the function a as an i6 I tupleWhere the ith coordinate is in Ai a aiieI amer az39 01139 pita The operation on H Ai was de ned by i6 Ma 395 Pz39a lmy 2367 1 39BW 1239 3950 i6 11 1395 Mi 39 ii617 that is aihd 39 bii61 ai 39bii 61
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