Fundamental Structures of Algebra and Discrete Mathematics Book

	Illustrations
	Preface
I	Sets	1
1	Elementary Constructions and Axioms	1
2	Cardinal and Ordinal Numbers	11
3	Intersections	21
II	Ordered Sets	31
1	Relations, Orders, and Zorn's Lemma	31
2	Lattices and Closures	43
3	Covering Relations	49
4	Intersecting Convex Sets	54
III	Groups	59
1	Binary Operations, Homomorphisms, and Congruences	59
2	Permutation Groups	68
3	Integers and Cyclic Groups	75
4	Alternating Groups	82
IV	Rings	93
1	Ideals	93
2	Polynomials	104
3	Factorization and the Euclidean Algorithm	111
V	Fields	125
1	Rational and Real Numbers	125
2	Galois Groups and Imaginary Roots	137
VI	Vector Spaces	157
1	Bases	157
2	Linear Maps and Equations	167
3	Affine and Projective Geometry	176
4	Hyperplanes in Linear Programming	183
5	Time and Speed in Special Relativity	187
VII	Graphs	197
1	Trees and Median Graphs	197
2	Games	202
3	Chromatic Polynomials	207
VIII	Lattices	213
1	Complements and Distributivity	213
2	Boolean Algebra	230
3	Modular and Geometric Lattices	237
IX	Matroids	245
1	Linear and Abstract Independence	245
2	Minors and Tutte Polynomials	252
3	Greedy Optimization Procedures	262
X	Topological Spaces	269
1	Filters	269
2	Closure, Convergence, and Continuity	272
3	Distances and Entourages	281
XI	Universal Algebras	291
1	Homomorphisms and Congruences	291
2	Algebra of Syntax	295
3	Truth and Formal Proof	300
XII	Categories	311
	Index of Definitions	327
	Index of Notation	339
	Index of Theorems	343