Algebra Over a Field Types

Home , Algebra over a field, Associative algebra, Euclidean domain, Exponential field, Flexible algebra, Free algebra

Riemann Manifold Types: A smooth manifold M 2 Functions: gp ∶ TpM → R Relations: Properties: gp is the inner product on the tangent space of a point p gp(X,Y ) = gp(Y,X) gp(aX + Y,Z) = agp(X,Z) + gp(Y,Z) M’s metric tensor is now positive deﬁnate

Forest Any point on M Pseudo Riemann Manifold has a positive def- Types: An undirected graph G = (V,E) Types: A smooth manifold M Functions: inite metric tensor 2 Functions: gp ∶ TpM → R Relations: Relations: every point Properties: G’s connected component are trees, but G need not be connected Properties: g is the inner product on the tangent space of a point p on M has a p g (X,Y ) = 0 ∀Y Ô⇒ X = 0 Smooth Manifold non degenerate p Types: A manifold M metric tensor Functions: Relations: A collec- Properties: Derivatives of arbitrary orders exist tion of trees

Line Graph Types: An undirected graph L(G) = (V ∗,E∗), with underlying undirected graph G = (V,E) Bipartite Graph M is infintiely ∗ Types: A k-partite graph G = (V,E) Functions: f ∶ E → V Edges represent Tree differentiable ∼∗= {{ } ∈ ∗}S Functions: Relations: x, y E vertces (and Types: A undirectd graph G = (V,E) ( ) Relations: Properties: f maps edges in G to vertexes in L G visa versa) of Functions: { ∗ ∗} ∈ ∗ Properties: Special case of k-partite graph for k = 2 An unordered pair of vertices vi , vj E iff their corresponding edges in G share an endpoint another graph, G Relations: Manifold V ,V are disjoint sets partition G’s vertices Properties: There are no cycles in G 1 2 Types: A topological space, M All e ∈ E can be written as {v , v } for some v ∈ V and v inV G is connected i i j i 1 j 2 Functions: G has no odd cycles Compact Manifold Relations: G is 2-colorable M is compact ?? Types: A manifold M Properties: ∀p ∈ M, ∃u ∈ U(p) ∶ u ≅ n as a space R A special vertex Functions: Manifolds are not inherently metric or inner product spaces is called the root Relations: each point locally Properties: X = ⋃x∈C x, then for F ⊆ C,F finite, X = ⋃x∈F x ’looks like’ G can be prti- euclidean space tioned into Rooted Tree G is acyclic Hypercube Graph exactly 2 sets Types: A tree G and connected Types: A regular, bipartite graph Qn = (X,E) Functions: Functions: Relations: Relations: n n−1 Properties: One vertex is designated as the root Properties: Qn has 2 vertices and 2 n edges

K-partite Graph The children Types: A graph G = (V,E) Euclidean Space of each node G is directed Functions: Types: A banach and hilbert space n R are ordered Relations: Functions: Undirected graph H is the graph Properties: V ,V , ..., V are disjoint sets that partition G’s vertices Relations: Types: A graph G = (V,E) 1 2 n » » of an n dimen- All ei ∈ E can be written as {vi, vj} where vi ∈ Vi, vj ∈ Vj and i ≠ j n 2 Functions: Properties: A norm is» defined as YxY = ⟨x, x⟩ = ∑i=1(xi) Ordered Tree Arboresence and Antiarboresence G is directed sional cube G is k-colorable n 2 Relations: d(x, y) = ∑i=1(xi − yi) Types: A directed graph G = (V,E) Types: A rooted polytree, G = (V,E) Functions: Functions: Properties: ∼ is irreflexive and symmetric Relations: Relations: Endow with an Endow with Properties: The children of nodes in G are ordered Properties: Every node has a directed path away from or toward the root inner product a norm Regular Graph Types: A graph G = (V,E) Functions: There is a directed Relations: path between Properties: All v ∈ V are connected to the same number of edges any vertex V can be di- Multigraph and the roote Hyper Graph vided into k Types: A tuple G = (V,E) Types: A tuple H = (V,E) pairs in E re- independant sets Functions: Functions: main unordered All vertexes have Relations: Relations: the same number More than 1 Properties: E is a multiset Hilbert Space Properties: E = {eiSi ∈ Ie, ei ⊆ V } Banach Space Polytree of neighbors edge can exist More than one edge may connect the same two vertices Types: An inner product space H Ie is an index set for edges in H Types: A vector space V Types: A directed graph G = (V,E) between the Functions: Functions: Functions: same two nodes Relations: Relations: » Relations: Properties: A norm may be defined as YxY = ⟨x, x⟩ Properties: for any Cauchy Sequence {xn}, limx→∞Yxn − xY = 0 Properties: G’s underlying undirected graph is a tree for any Cauchy Squence {xn}, limx→∞Yxn − xY = 0 G is acyclic and connected Chordal Graph Types: A perfectly orderable graph G = (V,E) Archimedian Local Field Nonarchimedian Local Field elements in E may Functions: Types: A local field K Types: A local field K V is complete V is complete be any subset of V Relations: as a space as a space Functions: Functions: Properties: All induced cycles are of order 3 Relations: Relations: The manifold G has a perfect elimination order Properties: The sequence SnS is unbounded Properties: The sequence SnS is bounded Any cycke of 4 or additionally has n≥1 n≥1 Simple Graph Perfect Graph more has a chord group structure Simple Directed Graph Types: A tuple G = (V,E) Types: A graph G = (X,E) Perfectly Orderable Graphs Chromatic number Inner Product Space Types: A graph G = (V,E) Functions: Functions: ω ∶ G ↦ n Types: A perfect graph G = (V,E) Normed Vector Space of every induced Types: A vector space V and field of scalars F Functions: pairs in E Relations: ∼= {{x, y} ∈ V × V S x, y are ajecent vertices in G} χ ∶ G ↦ n Functions: Types: A metric and topological vector space V 2 subgraph is G has a relation, < Functions: ⟨⋅, ⋅⟩ ∶ V → F Absolute Relations: = {(x, y) ∈ V × V S(x, y) ∈ E} are ordered Properties: V is the set of vertices in G Relations: Relations: <= {(x, y) ∈ V × V S x < y} Functions: YY ∶ V → R the size of the Relations: Absolute Value is a Properties: is irreflexive E is made of two element subsets of V Properties: ω(G) is the the size of the largest clique in G Properties: < is a total order Relations: Value is not a endowed with largest clique Properties: ⟨x, y⟩ = ⟨y, x⟩ bounded function ∀u, v ∈ V, u v ⇏ v u ∼ is the adjacency relation χ(X) is the the chromatic number of G For any chordless path abcd ∈ G and a < b, then d ≮ c Properties: YxY > 0 for x ≠ 0 and YxY = 0 iff x = 0 bounded function Graph Algebra a function to edges represent ⟨x, x⟩ ≥ 0 ∼ is irreflexive and symmetric ω(G) = χ(G) YαxY = SαSYxY Types: A directed graph D = (V,E) multiply vertices, ⋅ comparable Comparability Graph ⟨x, x⟩ = 0 iff x = 0 2 Yx + yY ≤ YxY + YyY Functions: ⋅ ∶ V → V ⋃ {0} elements in a poset Types: A perfect graph C = (S, ⊥) ⟨ax, y⟩ = a⟨x, y⟩ YY induces a metric on V ∶ d(x, y) = Yx − yY Relations: There exists a Functions: ⟨x + y, z⟩ = ⟨x, z⟩ + ⟨y, z⟩ ⎧ Local Field ⎪x x, y ∈ V, (x, y) ∈ E path between Relations: ⊥= {(x, y) ∈ S S x < y} Properties: x ⋅ y ∶ ⎨ any two veritces Properties: S is the set of vertices Types: A topological Field K ⎪0 x, y ∈ V ⋃ {0}, (x, y) ∉ E ⎩ on the graph S is a poset under < Functions: SS ∶ K → R≥0 0 ∉ V ordered pairs Complete Metric Space ⊥ is the edge relation Relations: in E the struc- Types: A metric space M Topological Vector Space Properties: ∀u, v ∈ K, ∃U, V ∈ τ ∶ u ∈ U, v ∈ V, U ∩ V = ∅ (x, y), (y, z) ∈ E Ô⇒ (x, z) ∈ E Equipped with ture of a group Functions: Types: A vector space V over a topological field K if C ⊆ τ satisfies X = ⋃ x, then there exists F ⊆ C,F finite X = ⋃ x equipped with 2 ⟨ ⟩ x∈C x∈F Connected Graph Relations: Functions: + ∶ V → V a function , a function YY Types: A graph G = (V,E) ∀ ∀ ∶ ∃ ∈ ∶ × ⊆ k × V → V Properties: F U A F A A U, F converges Cayley Graph Functions: d(x, y) = Yx − yY −−1 ∶ V → V + Types: A directed graph Γ = Γ(G, S), S a generating set for group G Relations: Relations: Functions: f ∶ G → V (G) Each vertex has Properties: There exists a path between any two v ∈ V in the graph Properties: k × V represents scalar multiplication by elements in a field K K is a field of Each edge has a unique label −1 g ∶ S → CS +, −+ and scalar multiplication are continuous scalars for V Has an absolute Relations: E(Γ) = {(g, gs) ∈ G × G S ∀g ∈ G, ∀s ∈ S} a unique label value function Properties: f assigns each g ∈ G to a vertex in Γ g assigns each s ∈ S to a unique color M is complete as a add a mean- Topological Field The identity element generally is ignored, so the graph does not contain loops Vertex Labeled Graph Strict Totally Ordered Set topological space ingful origin Types: A field K with topology τ Types: A graph G = (V,E) Types: A poset P 2 Functions: ⋅ ∶ K → K ∶ (x, y) ↦ x ⋅ y Functions: f ∶ V → L Functions: 2 Edge Labeled Graph + ∶ K → K ∶ (x, y) ↦ x + y Relations: Relations: <= {(x, y) ∈ P × P S x < y} −1 Types: A graph G = (V,E) and set L −+ ∶ K → R Properties: f maps vertexes to a set of labels L Properties: < is irreflexive, transitive, asymmetric and trichotomous Metric Space −1 Functions: f ∶ E → L Affine Space −× ∶ K~{0} → K~{0} Types: A topological space M Ð→ Relations: Relations: Functions: d ∶ M 2 → Types: A set A and vector space A −1 −1 Properties: f maps edges to a set of labels L R Ð→ Properties: +, ⋅, −+ , −⋅ are continuous mappings Relations: Functions: A × A → A ∶ (a, v) ↦ a + v Finite Countable Set all elements in P ≤ becomes Properties: d(x, y) ≥ 0 Relations: Types: A set S Ð→ Functions: are comparable irreflexive d(x, y) = 0 ⇔ x = y Properties: A’s additive group acts freely and transitively on A ⋅ has inverses Relations: d(x, y) = d(y, x) Ð→ S is in bijec- Uncountable Set ∀v, w ∈ A, ∀a ∈ A, (a + w) + u = a + (w + u) in the set Properties: SSS < ℵ d(x, z) ≤ d(x, y) + d(y, z) Ð→ 0 tion with a edge labels are ∀a, b ∈ A, ∃v ∈ R ∶ b = a + v Types: A set S Ð→ subset of the Functions: comparable ∀a ∈ A, A → A ∶ v ↦ a + v is bijective natural numbers Relations: Properties: SSS > ℵ0 Strict Poset Totally Ordered Set Closed sets are Types: A poset P Types: A poset P those who are the Countable Set Weighted Graph Functions: Functions: common zeros of Relations: <= {(x, y) ∈ P 2 S x < y} Relations: Topological Ring Types: A set S There is no in- Types: An edge labeled graph G = (V,E) and labels L a subset of the Properties: < is irreflexive, transitive, and asymmetric Properties: All elements are comparable under ≤ Types: A ring R with topology τ Functions: jection from S Functions: indeterminants 2 Functions: ⋅ ∶ R → R ∶ (x, y) ↦ x ⋅ y Relations: into the nat- Relations: in a polynomial 2 Closeness relation + ∶ R → R ∶ (x, y) ↦ x + y Properties: SSS ≤ ℵ0 ural numbers Properties: L is an ordered set ring over the same −1 Φ is replaced by −+ ∶ R → R underlying field Complete Uniform Space distance fucntion d Relations: Types: A uniform Space U Properties: ⋅ distributes over + −1 Infinite Countable Set S is in bijec- all elements in P Functions: +, ⋅, −+ are continuous mappings ≤ is now irreflexive Relations: Types: A set S tion with the are comparable Properties: ∀F ∀U ∶ ∃A ∈ F ∶ A × A ⊆ U, F converges Functions: natural numbers Relations: Graded Poset Properties: SSS = ℵ Partially Ordered Space 0 Types: A poset P bijection into S has an injec- Types: A poset X endowed with topology τ U is complete as a N Functions: ρ ∶ P → tion into the Functions: N topological space Relations: ⋖= {(x, y) ∈ P S ∄z ∶ x < z < y natural numbers Relations: ≤= {(x, y) ∈ X × XSx ≤ y} Properties: x < y Ô⇒ ρ(x) < ρ(y) Uniform Space add a new opera- Properties: ∀x, y ∈ X, x ≰ y ∶ ∃U, V ⊂ τ, x ∈ U, y ∈ V ∶ u ≰ v, ∀u ∈ U, ∀v ∈ V x ⋖ y Ô⇒ ρ(y) = ρ(x) + 1 Types: A topological space U and entourages Φ Topological Space tion + so (G, +) Functions: Types: A set X and τ ⊆ ℘(X) is a group and Relations: U = {(x, y)Sx ≈ y} Functions: (G, ⋅) is a moniod P endowed u Relations in Φ P has a rank Properties: Φ is a nonempty collection of relations on S Relations: Lie Group with a topology define closeness function, ρ(x) U ∈ Φ and U ⊆ V ⊆ X × X Ô⇒ V ∈ Φ Properties: X, ∅ ∈ τ Types: A topological group G that respects ≤ ∀U ∈ Φ, ∃V ∈ Φ, ∀x, y, z ∶ V ○ V ⊆ Φ ∶ x ≈V y, y ≈V z Ô⇒ x ≈U z xi ∈ τ Ô⇒ ⋂i∈I xi ∈ τ for finite I X has a group Functions: −1 ∀ ∈ ∃ ∈ ∀ ∶ ○ ⊆ ∶ ≈ Ô⇒ ≈ ∈ Ô⇒ ∈ Topological Group ⋅, − are smooth Relations: U Φ, V Φ, x, y V V Φ y V x x U y xi τ ⋃j∈J xj τ structure with con- Preordered Set Partially Ordered Set U, V ∈ Φ Ô⇒ U ∩ V ∈ Φ Types: A group G with topology τ maps, the under- Properties: ∀p ∈ G, ∃Up ≅ Br(p), for some r > 0 Set Locally Finite Poset tinuous functions Types: A set P Types: A set P Functions: ⋅ ∶ G2 → G lying topology is a −−1 and × are smooth maps Types: a collection of objects any subset of Types: A poset P equipped with Functions: Functions: −−1 ∶ G → G smooth manifold Functions: ⊂, ∩, ∪, ∖, ×, ℘, SS ≤ is antisymmetric P has finitely Functions: a preorder, ≤ Relations: ≤= {(x, y) ∈ P × P Sx ≤ y} Relations: ≤= {(x, y) ∈ P × P Sx ≤ y} 1 ∈ G Relations: ∈ many elements Relations: Properties: All elements need not be comparable under ≤ Properties: All elements need not be comparable under ≤ Relations: Properties: Properties: ∀x, y ∈ P, x ≤ y ∶ [x, y] has finitely many elements ≤ transitive and reflexive ≤ transitive, reflexive and antisymmteric Properties: ⋅, −−1 are continuous mappings P has group structure that respects ≤ each element is the greatest lower bound of S is equipped members of S are Neighborhoods of Partially Ordered group the compact with a relation → sets but S is not points must satisfy Types: A poset and group G elements below it stronger properties Functions: + ∶ G2 → G −1 −+ ∶ G → G 0 ∈ G Relations: ≤= {(x, y) ∈ G × G S 0 ≤ −x + y} any two ele- Convergent Space ≤ + ments have an Proper Class Properties: respects Types: A set S and filters on S, F Cardinal upper bound Algebraic Poset Types: a collection, C of sets The set of ordinals Functions: Types: Ordinals κ Types: A poset P Functions: ϕ(x) that cannot be put Relations: → Functions: Functions: Relations: in bijection with Upward (Downward) Linked Set Properties: ∀F ∈ F, x ∈ S, F → x ⇔ F convergences to X Relations: Relations: Properties: ϕ(x) is a membership function any lesser ordinal Types: A subset S of poset P → is centered, isotone, and directed Properties: The cardinality κ is the least ordinal, α, s.t. there is a bijection between S and α Properties: each element is the least upper bound of the compact elements below it C is not a set Functions: N (x) = ⋂{F ∈ FSF → x} Relations: Properties: ∀x, y ∈ S ∶ ∃z ∈ P s.t. x ≤ (≥)z and y ≤ (≥)z

Each element any two elements is either the set of P have an Magma of the elements upper bound Types: A set M before it or 0 Functions: ⋅ ∶ M 2 → M Relations: M is cancellitive ⋅ is associative Upwards (Downwards) Centered Set Properties: ∀u, v ∈ M ∶ u ⋅ v ∈ M Ordinal Types: A linked subset S of poset P Types: A collection of sets, Xi Functions: Quotient Ring n Functions: ×, +,X ,S(n) Relations: Types: A ring Q Quasigroup Relations: > Functions: Properties: ∀Z ⊆ P,Z has an upper (lower) bound ∈ P Semigroup Types: A magma Q Properties: Xk = {0, 1, 2, ...Xk−1} Relations: ∼= {(x, y} ∈ R × R S x − y ∈ I} −1 Prime Ideal ⋅, − are contin- Types: A magma S of elements Functions: Ó∶ Q2 → Q > is trichotomous, transitive, and wellfounded Properties: Q is constructed by dividing R with one of its ideals 2 Types: A subgroup of elements I uous mappings Functions: Ò∶ Q → Q Relations: 1R, 1L ∈ Q any subset Functions: S is endowed with Properties: ⋅ is associative Relations: of P has an Relations: Semiring a new operation, Properties: Q is a cancellative magma upper bound Properties: ab ∈ I Ô⇒ a ∈ I or b ∈ I Types: A set R + such that ⋅ Functions: ⋅ ∶ R2 → R distributes over + 2 Q has an iden- +R → R Q has an iden- 0 ∈ R tity element, 1 If the product of tity element, 1 Upward (Downward) Directed Set A ring formed Relations: two elements is Types: A preset P by the quotient Properties: + is a commutative monoid in I, then one of Functions: of a ring and ⋅ is a monoid and distributes over + Monoid Loop Maximal ideal those−1 elements Relations: one of its ideals +, −+ , ⋅ are Types: A semigroup S Types: A quasigroup Q Types: A subgroup of elements, I continuousmust be in I Principal ideal Functions: 1 ∈ S Functions: 1 ∈ Q Properties: ≤ is a preorder Functions: Types: A subgroup of elements, I Relations: Relations: ∀x, y ∈ P ∶ ∃z s.t. x ≤ (≥)z and y ≤ (≥)z Relations: Functions: P Group Properties: 1 is the identity for ⋅ Properties: 1 is the identity for ⋅ Properties: I ⊆ J Ô⇒ I = J or J = R Relations: Types: A group G 1 ⋅ u = u ⋅ 1 = u 1 ⋅ x = x ⋅ 1 = x Properties: I is generated by a single element Functions: Relations: I is not the proper Properties: ∀g ∈ G ∶ SgS = pn, p prime, n ∈ subset of any N other ideals of R I is generated from a single element Torsion Free Group Ideal any pair of any pair of all elements Types: A group G Types: A subgroup of a ring, I elements have a elements have in S have an ⋅ is associative Functions: Functions: unique greatest a unique least inverse in S Relations: Relations: n lower bound upper bound I is a right and left Properties: ∄g ∈ G, g ≠ 1 ∶ g = 1, n ∈ N Free Abelian Group Left (right) Ideal Properties: The left and right ideals generated by a subgroup are equal idea of the same all elements Types: A group F Types: A subgroup of a ring, I ∀x ∈ I, ∀r ∈ R ∶ rx ∈ I n underlying subset have order p Functions: Functions: Relations: Relations: there are no add commuta- Properties: F is generated by a set S Properties: ∀x ∈ I, ∀r ∈ R ∶ rx (xr) ∈ I elements in G tivity relation The only relation on F is commutativity Sigma Algebra of finite order Types: A subset Σ, of ℘(X), X a set Free Group A subset of the Topological group Functions: ∩, ∪, ∖ Group Types: A group F ?? ?? ring closed under Types:AbelianA Group group G Relations: Types: A set G Functions: Types: A poset P Types: A poset P left (right) ring Properties: X ∈ Σ Functions:Types: A group G ⋅ ∶ 2 → Functions: G G There are no fur- Relations: Functions: Functions: multiplication ∩, ∪ closed under countable operations Relations:Functions: −−1 ∶ G → G −1 ⋅ is commutative ther relations on G Properties: F is generated by a set S Relations: ≤= {(x, y) ∈ P × P S x ≤ y} Relations: ≤= {(x, y) ∈ P × P S x ≤ y} (R, +) be- Properties:Relations:− and × are continuous maps ∈ A subset of the 1 G There are no relations on FS (beyond group axioms) Properties: ∀x, y ∈ S, ∃c ∶ c ≤ x, c ≤ y Properties: ∀x, y ∈ P, ∃c ∶ x ≤ c, y ≤ c comes a group Properties: ⋅ is commutative ring closed under Relations: c is the unique greatest lower bound c is the unique least upper bound ring multiplication A ring under Properties: ⋅ is associative symmetric 1 is the identity for ⋅ Meet Semilattice: Algebraic Edition Join Semilattice: Algebraic Edition difference as + and −−1 is the inverse map for ⋅ G has a fi- Types: A poset P Types: A poset P −1 −1 nite number Functions: ∧ ∶ P 2 → P Functions: ∨ ∶ P 2 → P +, ⋅, −+ , −⋅ set intersection as ⋅ of elements Relations: Relations: Noncommutative Ring are continuous mappings G has no nor- Properties: ∧ is associative, commutative, and idempotent Properties: ∨ is associative, commutative, and idempotent Types: A ring R mal subgroups Functions: Relations: Properties: ⋅ need not be commutative Boolean Ring any pair of Artinian Ring Simple Group elements have Types: A commutative ring R Types: A ring R Commutative Ring Types: A group G a unique least All elemets are Functions: Functions: Functions: upper bound Types: A ring R Relations: G has a fi- Finite Group Relations: idempotent with Relations: Functions: respect to ⋅, + Properties: ∀x ∈ R ∶ x2 = x nite number Types: A group G Permutation Group Properties: For any I0 ⊇ I1 ⊇ ... ⊇ Ik there is some k ≥ 0 s.t. Ik = Ik+1 Relations: Properties: N ◁ G Ô⇒ N = {1} or G of generators ∀x ∈ R ∶ 2x = 0 Functions: Types: A group Pn Properties: ⋅ is commutative Relations: Functions: Properties: SGS = n, n ∈ N Relations: Properties: Pn < Sn ⋅ need not be G is closed under Pn contains only commutative multiplication by even permutations a ring of scalars Semimodule Finitely Generated Group A proper subset Noetherian Ring Finitely Presented Group Types: A set M Types: A group G G is the group of the Finite Types: A ring R + ∶ 2 → Types: A finitely generated group G Functions: M M Functions: G has fi- of all bijective Symmetric Group Functions: × → Functions: R M M Relations: nite relators functions from Alternating Group Relations: Ideals in R satisfy Relations: Relations: Properties: G is of the form ⟨SSR⟩, S a generating set, R relators the set to itself Types: A permutation group An Properties: For any I0 ⊆ I1 ⊆ ... ⊆ Ik there is some k ≥ 0 s.t. Ik = Ik+1 the descending Properties: + is associative and commutative Properties: SRS = n, n ∈ N SSS = n, n ∈ N Fintie Symmteric Group Functions: chain condition (M, +) is an commutative monoid Types: A finite group S Relations: ⋅ R × M is scalar multiplication by elements in a ring n is additionally Functions: Properties: An = {σSσ ∈ Sn ∧ σ even } Scalar multiplication is associative and distributes over addition commutative Relations: Ideals in R satisfy Properties: Sn = {σSσ ∶ X → X is bijective } the acending M is a chain condition Ring abelian group No nontriv- Types: A set R ial partitions Functions: ⋅ ∶ R2 → R Artinian Module G has one are preserved + ∶ R2 → R Types: A module M generator There are no −1 −+ ∶ R → R Functions: zero divsors in R Primitive Group 0 ∈ R Relations: Types: A permutation group G that acts on a set X Relations: Properties: For any chain of submodules S0 ⊇ S1 ⊇ ... ⊇ Sk there is some k ≥ 0 s.t. Sk = Sk+1 Domain Functions: Properties: (R, +) is an abelian group Relations: Types: A ring R M satisfies ⋅ is a monoid and distributes over + R is a ring of Properties: G preserves only trivial partitions Functions: decending chain scalars for an S is also a join S is also a meet Relations: condition on abelian group M Module semilattice semilattice ⋅ is commutative Properties: ∀a, b ∈ R, ab = 0 Ô⇒ a = 0 or b = 0 submodules Cyclic Group Types: A set M and obeys ab- and obeys ab- Semimodular Lattice 2 Types: A group G with element a Functions: + ∶ M → M Distributive Lattice sorption law sorption law Types: A lattice L −1 Functions: Modular Lattice Noetherian Module −+ ∶ M → M Cyclically Ordered Group G has a chain of Polycyclic Group Types: A modular lattice L L has a re- L’s functions Types: A semimodular lattice L Functions: M satisfies R × M → R Add a rela- Relations: cyclic subgroup Functions: stricted asso- Types: A module M Types: A group C Types: A group P with subgroups Hi distribute over Functions: Relations: ⋖= {(x, y) ∈ P S ∄z s.t. x < z < y} accending chain Relations: tion, [ ,, ] Properties: G can be generated from a and a set of relations R whose quotients Relations: ciative property Functions: Functions: Functions: each other Relations: Properties: a ∧ b ⋖ a Ô⇒ b ⋖ a ∨ b condition on Properties: + is associative and commutative G is necessarily commutative are also cyclic Properties: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) Relations: Relations: [ ,, ] = {(a, b, c) ∈ C × C × CS[a, b, c]} Relations: Properties: a ≤ c Ô⇒ a ∨ (b ∧ c) = (a ∨ b) ∧ c submodules ( +) ∨ ( ∧ ) = ( ∨ ) ∧ ( ∨ ) Properties: For any chain of submodules S0 ⊆ S1 ⊆ ... ⊆ Sk there is some k ≥ 0 s.t. Sk = Sk+1 M, is an abelian group Properties: [a, b, c] is cyclic, asymmetric, Properties: P is necessarily finitely presented x y z x y x z R × M is scalar multiplication by elements in a ring L is addition- Integral Domain All submodules of M are finitely generated transitive and total {Hi}i∈{1,2,..,n−1} ∶ Hi~Hi+1 is cyclic Scalar multiplication is associative and distributes over addition All finitely gen- G is a subgroup in ally algebraic Types: A domain R erated subgroups another group V and atomic Functions: of G are cyclic Relations: Ð→ Geometric Lattice Properties: ⋅ is additionally commutative n ≤ 2 Add a functon, M’s ring is differ- Types: A semimodular, algebraic, atmoic lattice L ential operators Heyting Algebra Functions: Types: A distributive lattice H Relations: 2 Virtually Cyclic Group Functions: Ð→∶ H → H Properties: L is finite Locally Cyclic Group Metacyclic Group Types: A group V with subgroup H Relations: Types: A group L Types: A group G Ring addition is Functions: Properties: (c ∧ a) ≤ b ⇔ c ≤ (a Ð→ b) All elements Functions: Functions: the symmetric Relations: Monoid (H, ⋅, 1) has operation ⋅ = ∧ can be factored Relations: Relations: If a covers the difference and Properties: ∃H ≤ V ∶ H is cyclic into the product Properties: ∀H ≤ L, if H is finitely generated, it is cyclic Properties: n ≤ 2 greatest lower L is additionally M has no ring multiplica- SV ∶ HS = n, n ∈ N algebraic and of a unit and M is generated bound of a and zero divisors tion is greatest Atomic Lattice semimodular prime elements D Module from a finite set b, then b is lower bound a ⋅ b = a ∧ b Types: A lattice L Types: A module M[X] covered by the Functions: Functions: ∂ ∶ M → M greatest lower Ri Relations: ⋖= {(x, y) ∈ P S ∄z ∶ x < z < y} Relations: bound of a and b Finitely Generated Module Properties: ∀b ∈ L ∶ 0 ∧ b = 0 L is additionally Properties: X = {x , x , ..., x } where x is an indeterminant H’s underlying 1 2 n i Types: A module M ∃a ∈ L ∶ 0 ⋖ a semimodular ∂ is in the ring of scalar multiplication lattice is complete i i Ri Functions: ∀b ≠ a , ∃a ∶ 0 < a < b and atomic ∂ satisfies the Leibniz product rule i i i Ri Relations: { } M’s generating Simple Module ai i∈I is the set of atoms in L Properties: M has a finite number of generators set is one element Residuated Lattice Unique Factorization Domain Types: A module M Types: A lattice L Types: An integral domain R Torsion Free Module Cyclic Module Functions: Complete Heyting Algebra M has no proper Functions: ⋅ ∶ L2 → L Functions: Types: A module M Types: A finitely generated module M Relations: Types: A heyting algebra H submodules 1 ∈ L Relations: Functions: Functions: Properties: The only submodules of M are 0 and M Functions: L only must satisfy Relations: Properties: All ideals are finitely generated Relations: Relations: Relations: the Modular law Properties: (L, ⋅, 1) is a monoid All irreducible elements are prime Properties: ∄m ∈ M, r, m ≠ 0 ∶ r ⋅ m = 0 Properties: M is generated from a single element Properties: H is complete as a lattice in the special (L, ≤) is a lattice ⊥ ∀x ∈ R, x ≠ 0, x = up p ...p for u unit, p prime case b = a 1 2 n i ∀z ∶ ∀x, there exists a greatest y and ∀y, there exists a greatest x s.t. x ⋅ y ≤ z All non zero elements are either Algebraic Lattice atoms or compa- Types: A continuous lattice A rable to atoms Functions: Relations: Properties: Each element is the least of compact elements below it: those x who satisfy x ≪ a All ideals are principal Lattice (order theory) Types: A set L Functions: Ð→= ¬x ∨ y L has an element each are the great- 1 and a function ⋅ Relations: ≤= {(x, y) ∈ L × S x ≤ y} est lower bound Principal Ideal Domain such that (L, ⋅, 1) Properties: ∀x, y ∈ L, ∃c, d ∶ c is the greatest lower bound and d is the least upper bound of the compact Types: A unique factorization domain R is a monoid elements below it Functions: Relations: Properties: All ideals are principal Any two elements have a greatet common divisor Relational Algebra M has no re- Types: A commutative algebra R Complete Lattice lators beyind Functions: ∨ ∶ R2 → R Types: A lattice L module axioms ∧ ∶ R2 → R Functions: − ∶ R → R Relations: Relations: Properties: ∀A ⊆ L, A has a greatest lower bound and least upper bound Properties: Lattice (Algebra) R can be en- Types: A set L dowed with a eu- Functions: ∨ ∶ L2 → L 2 clidean algorithm Orthomodular Lattice ∧ ∶ L → L Relations: Types: A lattice L a is the least upper Properties: (L, ∨), (L, ∧) are both semilattices connected by absorption laws Lie Algebra Functions: bound for the ∀a, b ∈ L ∶ a ∨ (a ∧ b) = a, a ∧ (a ∨ b) = a Euclidean Domain Types: A nonassociative Algebra g :( Relations: set of elements Exponential Field Types: A principal idea domain R Functions: [] ∶ g2 → g Properties: a ≤ b Ô⇒ b = a ∨ (b ∧ a⊥) way below a Types: A field K Functions: f ∶ R ∖ {0} → R Relations: i Functions: Φ ∶ K+ → K× Relations: Properties: ∀x, y ∈ g ∶ [x, y] = −[y, x] Relations: Properties: f is the euclidean (gcf) algorithim ∀x, y, z ∈ g ∶ [x, [y, z]] + [y[z, x]] + [z, [x, y]] = 0 Polynomial Algebra i Properties: Φ is a homomorphism ∀ ∈ ∶ [ + ] = [ ] + [ ] [ + ] = [ ] + [ ] R may have many fi or just one Differential Field x, y, z g ax by, z a x, z b y, z and x, ay bz a x, y b x, z Types: A commutative Algebra A[X] Free Module [x, x] = 0 Functions: 1 ∈ A[X] Types: A polynomial field K every subset in Continuous Lattice The euclidean Types: A projective module M Relations: Functions: ∂i ∶ K → K L has a least Types: A complete lattice A algorithm is one there exists a Functions: Properties: X = {x , x , ...x } where x is an indeterminate, n ≥ 1 Boolean Algebra L has a greatest Relations: 1 2 n i upper bound Functions: for all values homomorphism Relations: Types: A complemented, distributive lattice and algebra B lower bound and Properties: Finitely many ∂ exist over K All elements and a greatest Relations: ≪= {(x, y) ∈ L × L S ∀D ⊆ L ∶ y ≤ sup D, ∃x ∈ L, ∃d ∈ D s.t. x ≤ d} in the field from the addivitive i Properties: M has no further relations beyond module axioms Functions: ¬ ∶ B → B least upper bound K has finitely ∂i satisfies the Leibniz product rule 2 have relative lower bound Properties: ∀x ∈ P, {a S a ≪ x} is directed and has least upper bound x to multiplica- + M has a basis ⊕ ∶ B → B L satisfies a ∨ (a⊥ ∧ tive groups many dera- ∂i is a homomorphism of K eqippued with A has an alphabet complements Flexible Algebra Relations: c = c for a ≤ c vations on it a function [ , ] of indeterminates 2 Types: A nonasscoative Algebra A Properties: ∀x ∈ B, x = x Functions: ¬ is negation (complementation) Relations: x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y) ⊕ is algebra addition, ∧ is algebra multiplication L additionally x(yx) = (xy)x ∀ ∈ ∶ ( ) = ( ) ′ ′ Properties: x, y A x yx xy x satisfies (x ∧ y ) = Relatively Complemented Lattice All elements y ∨ (x′ ∧ y′) Types: A lattice L are idempotent Functions: Relations: Field Power Associative Algebra Properties: ∀c∀d ≥ c, [c, d] ∶ ∀a ∈ [c, d], ∃b s.t.: Types: A ring K Types: A nonassociative Algebra A ∨ = Commutative Algebra Orthocomplemented Lattice a b d Bounded Lattice Functions: + ∶ K2 → K Functions: a ∧ b = c Types: An algebra A Types: A lattice L Types: A lattice L Characteristic Zero Field The multiplica- × ∶ K2 → K Indeterminates Relations: x(x(xx)) = a and b are relative complements Nonassocative Algebra Alternative Algebra Functions: Functions: ⊥∶ L → L Functions: Types: A field K tive identity −−1 ∶ K → K in M are basis Properties: ∀x, y ∈ A ∶ x(x(xx)) = (xx)(xx) = (x(xx))x (xx)(xx) = + Types: An algebra A Types: A nonassocative Algebra A Relations: Relations: Relations: Functions: cannot be added −−1 ∶ K~{0} → K~{0} vectors for V (x(xx))x × Functions: xx(y) = x(xy) Functions: Properties: ⋅ is commutative Properties: ⊥ maps an element a to its complement Properties: ∀x ∈ L ∶ Relations: finitely many 0, 1 ∈ K Relations: Relations: A eqipped with a ∨ a⊥ = 1 x ∧ 1 = x Properties: ∄n ∈ ∶ 1 + 1 + ... + 1 = 0 times tog et the Relations: N Properties: ⋅ is need not be associative Properties: ∀x, y ∈ A ∶ xx(y) = x(xy) finitely many a ∧ a⊥ = 0 x ∨ 1 = 1 ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ additive identity Properties: +, × both associate and commute for each element n derivations (a⊥)⊥ = a L itself can act x ∨ 0 = x × distributes over + in L, there is a a ≤ b Ô⇒ a⊥ ≥ b⊥ as the interval x ∧ 0 = 0 + × 0 is the identity for and 1 is the identity for K acts as scalars second element equipped with a ( +) ( ×) Complements K, and K, are groups over a free module in L such that partial order, ≤ ⋅ is commutative Differential Algebra are unique glb and the lup ⋅ need not be Types: A commutative algebra A are the maxium assocative Functions: ∂i ∶ A → A Complemented Lattice and minimum Ordered Field Relations: Types: A lattice L element of L Types: A characteristic zero field K Properties: There are finitely many ∂i over A Functions: Functions: All ∂ satisfy the Leibniz product rule (xy)(xx) = i Relations: Relations: ≤ ∂ is a homomorphism of A+ ( ( ( )) i Properties: ∀a ∈ L, ∃b ∈ L ∶ Properties: ≤ respects + and × Vector Space x y xx a ∨ b = 1 K is necessarily infinite Types: A set V over a field K Jordan Algebra 2 a ∧ b = 0 The multiplica- Functions: + ∶ V → V Types: A nonassociative algebra A Extension such −1 tive identity −+ ∶ V → V Functions: can be added that a polymonial K × V → V canbe decompased Relations: finitely many 0 ∈ V Properties: ∀x, y ∈ A ∶ (xy)(xx) = (x(y(xx)) times tog et the into liner factors Relations: additive identity Properties: + is associative and commutative Zero Algebra Characteristic Nonzero Field K × V is scalar multiplication by K Types: An algebra A Types: A field K (V, +) is a group Algebra Over a Field Functions: Relations: Functions: Splitting Field Types: A set of elements A, with underlying field K 2 Properties: ∀u, v ∈ A, uv = 0 Relations: Types: A field K′, along with field K, polynomial ring K[X] Functions: ⋅ ∶ A → A the product of all Properties: ∃n ∈ ∶ 1 + 1 + ... + 1 = 0 + ∶ A2 → A A is inherently associative and commutative N ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ Functions: elements in A is 0 n Relations: K × A → A Properties: K′ is a algebraic extension of K Relations: K′ of some x ∈ K[X] is the smallest extension so that x can be decomposed into linear factors Properties: + is associative and commutative ⋅ is not assumed commutative or associative ⋅ distributes over + ⋅ is associative equipped with a A map from a ≤ K × A represents scalar multiplication partial order, g to a unital (A, +,K) is a vector space All polynomials assocative algebra can be decompased ⋅ need not be into linear factors commutative

?? Types: An associative algebra and locally finite poset, A Finite Field Algebraically Closed Field Noncommutative Algebra Associative Algebra Functions: δ ∶ A → A Types: A characteristic nonzero field K Types: A field K and a polynomial ring K[X] A made up of Types: An algebra A Types: An algebra A ∗ ∶ A2 → A Functions: Functions: functions that Functions: Functions: Relations: Relations: Relations: assign subinte- Relations: Relations: Properties: ∗ is algebra multiplication Properties: K has a finite number of elements Properties: All polynomials x ∈ K[X] can be decomposed as the product of linear factors vals to scalars Properties: ⋅ in A is need not be commutative Properties: ⋅ is associative Any a ∈ A behaves such that [b, c] ↦ a(b, c) The order of some k ∈ K is pn for some p prime, n ∈ Equivalently, there is no proper algebraic extension of K A has an alphabet N The idenitity a(b, c) is taken from a commutative ring of scalars Free Algebra of indeterminates for ⋅ is in A δ is the identity function Types: A noncommutative algebra A[X] Functions: 1 ∈ A[X] The symmetric Relations: algebra is a free Properties: X = {x1, x2, ..., xn}, xi is an indeterminate unital commu- 1 is the empty word tative algebra Unital Algebra Members of A are Types: An algebra A linear opertors Functions: 1 ∈ A T (V ) is the free Relations: Operator Algebra A equipped algebra over a Properties: 1 is the identity for ⋅ Types: An associative Algebra A with a norm, YY vector space Functions: ○ ∶ A2 → A ⋅ is the ten- Relations: sor product Properties: a ∈ A are continuous linear operators over a topological vector space ○ is operator composition (and algebra multiplication) Tensor Algebra Banach Algebra Types: A noncommutative Algebra T Types: An associative algebra A Functions: ⊕ ∶ T 2 → T Functions: SS SS ∶ A → R ⊗ ∶ T 2 → T Relations: Relations: Properties: ∀x, y ∈ A ∶ SSxySS ≤ SSxSS SSySS Properties: T is defined over any vector space V A is also a Banach space k T V = V ⊗1 V ⊗2 ... ⊗k V T (V ) = C ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ .... T kV ⊗ T lV = T k+lV T (V ) is the most general algebra over V

⋀ is a quoient ⋀ is a quoient algebra of T (V ) algebra of T (V )

Exterior Algebra Universal Enveloping Algebra ⋁ is a quoient Types: A nonassociative algebra ⋀(V ) of a vector space V Types: A unital associative Algebra U(g) of a Lie Group g algebra of T (V ) Functions: Functions: Relations: Relations: Properties: I is the ideal generated from w ⊗ v + v ⊗ w, ∀w, v ∈ V Properties: I is the ideal generated from x ⊗ y − y ⊗ x − [x, y]∀x, y ∈ T (g) ⋀(V ) = T (V )~I U(g) = ⊗g)~I Division by the anticommutator makes ⋁(V ) anticommutative

U(g)’s underlying Lie Algebra is abelian

Symmetric Algebra Types: An associative, commutative algebra ⋁(V ) of a vector space V Functions: 1 ∈ ⋁(V ) Relations: Properties: I is the ideal generated from w ⊗ v − v ⊗ w, ∀w, v ∈ V ⋁(V ) = T (V )~I Division by the commutator makes ⋁(V ) commutative 2 ALGEBRA:COMMUTATIVE ALGEBRAS

Section 1 Algebra: Basic

Algebra Over a Field Types: A set of elements A, with underlying ﬁeld K Functions: ⋅ ∶ A2 → A + ∶ A2 → A K × A → A Relations: Properties: + is associative and commutative ⋅ is not assumed commutative or associative ⋅ distributes over + × by k ∈ K represents scalar multiplication (A, +,K) is a vector space not assumed commutative or associative ⋅ distributes over +

Section 2 Algebra:Commutative Algebras

Commutative Algebra Types: An algebra A Functions: Relations: Properties: ⋅ is commutative

Diﬀerential Algebra Types: A commutative algebra A Functions: ∂i ∶ A → A Relations: Properties: There are ﬁnitely many ∂i over A All ∂i satisfy the Leibniz product rule + ∂i is a homomorphism of A

2 3 ALGEBRA: ASSOCIATIVE ALGEBRA

Polynomial Algebra Types: A commutative Algebra A[X] Functions: 1 ∈ A[X] Relations: Properties: X = {x1, x2, ...xn} where xi is an indeterminate, n ≥ 1

Heyting Algebra Types: A distributive lattice H Functions: Ð→∶ H2 → H Relations: Properties: (c ∧ a) ≤ b ⇔ c ≤ (a Ð→ b) Monoid (H, ⋅, 1) has operation ⋅ = ∧

Boolean Algebra Types: A complemented, distributive lattice and algebra B Functions: ¬ ∶ B → B ⊕ ∶ B2 → B Relations: Properties: ∀x ∈ B, x2 = x ¬ is negation (complementation) x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y) ⊕ is algebra addition, ∧ is algebra multiplication

Relational Algebra Types: A commutative algebra R Functions: ∨ ∶ R2 → R ∧ ∶ R2 → R − ∶ R → R Relations: Properties:

Section 3 Algebra: Associative Algebra

3 4 ALGEBRA: NONCOMMUTATIVE ALGEBRA

Associative Algebra Types: An Algebra A Functions: Relations: Properties: ⋅ is associative

Banach Algebra Types: An associative algebra A Functions: SS SS ∶ A → R Relations: Properties: ∀x, y ∈ A ∶ SSxySS ≤ SSxSS SSySS A is also a Banach space

An associative algebra and locally ﬁnite poset, A δ ∶ A → A ∗ ∶ A2 → A ∗ is algebra multiplication Any a ∈ A behaves such that [b, c] ↦ a(b, c) a(b, c) is taken from a commutative ring of scalars δ is the identity function

Operator Algebra Types: An associative Algebra A Functions: ○ ∶ A2 → A Relations: Properties: a ∈ A are continuous linear operators over a topological vector space ○ is operator composition (and algebra multiplication)

Section 4 Algebra: Noncommutative Algebra

Noncommutative Algebra Types: An algebra A Functions: Relations: Properties: ⋅ in A is need not be commutative

4 4 ALGEBRA: NONCOMMUTATIVE ALGEBRA

Free Algebra Types: A noncommutative algebra A[X] Functions: 1 ∈ A[X] Relations: Properties: X = {x1, x2, ..., xn} where xi is an indeterminate from an alphabet 1 is the empty word

Tensor Algebra Types: A noncommutative Algebra T Functions: ⊕ ∶ T 2 → T ⊗ ∶ T 2 → T Relations: Properties: T is deﬁned over any vector space V k T V = V ⊗1 V ⊗2 ... ⊗k V T (V ) = C ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ .... T kV ⊗ T lV = T k+lV T (V ) is the most general algebra over V

Universal Enveloping Algebra Types: A unital associative Algebra U(g) of a Lie Group g Functions: Relations: Properties: I is the ideal generated from x ⊗ y − y ⊗ x − [x, y]∀x, y ∈ T (g) U(g) = ⊗g)~I

Exterior Algebra Types: A nonassociative algebra ⋀(V ) of a vector space V Functions: Relations: Properties: I is the ideal generated from w ⊗ v + v ⊗ w, ∀w, v ∈ V ⋀(V ) = T (V )~I Division by the anticommutator makes ⋁(V ) anticommutative

5 5 ALGEBRA: NONASSOCIATIVE ALGEBRA

Section 5 Algebra: Nonassociative Algebra

Nonassocative Algebra Types: An algebra A Functions: Relations: Properties: ⋅ is need not be associative

Jordan Algebra Types: A nonassociative algebra A Functions: Relations: Properties: ∀x, y ∈ A ∶ (xy)(xx) = (x(y(xx))

Power Associative Algebra Types: A nonassocative Algebra A Functions: Relations: Properties: ∀x, y ∈ A ∶ x(x(xx)) = (xx)(xx) = (x(xx))x

Flexible Algebra Types: A nonassociative Algebra A Functions: Relations: Properties: ∀x, y ∈ A ∶ x(yx) = (xy)x

Alternative Algebra Types: A nonassociative Algebra A Functions: Relations: Properties: ∀x, y ∈ A ∶ xx(y) = x(xy)

6 6 ALGEBRA: NOT FURTHER CATEGORIZED

Lie Algebra Types: A nonassociative Algebra g Functions: [] ∶ g2 → g Relations: Properties: ∀x, y ∈ g ∶ [x, y] = −[y, x] ∀x, y, z ∈ g ∶ [x, [y, z]] + [y[z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ g ∶ [ax + by, z] = a[x, z] + b[y, z] and [x, ay + bz] = a[x, y] + b[x, z] [x, x] = 0

Section 6 Algebra: Not further categorized

Zero Algebra Types: An algebra A Functions: Relations: Properties: ∀u, v ∈ A, uv = 0 A is inherently associative and commutative

Unital Algebra Types: An algebra A Functions: 1 ∈ A Relations: Properties: 1 is the identity for ⋅

7 8 FIELDS: TOPOLOGICAL + RELATED

Section 7 Fields: Basic

Field Types: A ring K Functions: + ∶ K2 → K × ∶ K2 → K −1 −+ ∶ K → K −1 −× ∶ K~{0} → K~{0} 0, 1 ∈ K Relations: Properties: +, × both associate and commute × distributes over + 0 is the identity for + and 1 is the identity for × (K, +) and (K, ×) are groups

Section 8 Fields: Topological + related

Topological Field Types: A topological ring F Functions: ⋅ ∶ R2 → R ∶ (x, y) ↦ x ⋅ y + ∶ R2 → R ∶ (x, y) ↦ x + y −1 −+ ∶ R → R −1 −× ∶ F ~{0} → F ~{0} Relations: −1 −1 Properties: +,⋅, −+ , −× are continuous mappings

Local Field Types: A topological Field K Functions: SS ∶ K → R≥0 Relations: Properties: ∀u, v ∈ K, ∃U, V ∈ τ ∶ u ∈ U, v ∈ V, U ∩ V = ∅ if C ⊆ τ satisﬁes X = ⋃x∈C x, then there exists F ⊆ C,F ﬁnite X = ⋃x∈F x

8 9 FIELDS: CHARACTERISTICS

Archimedian Local Field Types: A local ﬁeld K Functions: Relations: Properties: The sequence SnSn≥1 is unbounded

Nonarchimedian Local Field Types: A local ﬁeld K Functions: Relations: Properties: The sequence SnSn≥1 is bounded

Section 9 Fields: Characteristics

Characteristic Zero Field Types: A ﬁeld K Functions: Relations: Properties: ∄n ∈ ∶ 1 + 1 + ... + 1 = 0 N ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n

Ordered Field Types: A characteristic zero ﬁeld K Functions: Relations: ≤ Properties: ≤ respects + and × K is necessarily inﬁnite

Characteristic Nonzero Field Types: A ﬁeld K Functions: Relations: Properties: ∃n ∈ ∶ 1 + 1 + ... + 1 = 0 N ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n

9 10 FIELDS: NOT FURTHER CATEGORIZED

Finite Field Types: A characteristic nonzero ﬁeld K Functions: Relations: Properties: K has a ﬁnite number of elements n The order of some k ∈ K is p for some p prime, n ∈ N

Section 10 Fields: Not further categorized

Vector Space Types: A set V over a ﬁeld K Functions: + ∶ V 2 → V −1 −+ ∶ V → V K × V → V 0 ∈ V Relations: Properties: + is associative and commutative K × V is scalar multiplication by K (V, +) is a group

Exponential Field Types: A ﬁeld K Functions: Φ ∶ K+ → K× Relations: Properties: Φ is a homomorphism

Differential Field Types: A polynomial field K Functions: ∂i ∶ K → K Relations: Properties: Finitely many ∂i exist over K ∂i satisfies the Leibniz product rule + ∂i is a homomorphism of K

10 12 GRAPHS: BASIC

Section 11 Fields: Algebraic

Splitting Field Types: A polynomial ﬁeld K Functions: Relations: Properties: K is extended so that some polynomial (or set of polynomials) k ∈ K can be written as the product n linear factors, where n is the degree of the polynomial

Algebraically Closed Field Types: A ﬁeld K and a polynomial ring K[X] Functions: Relations: Properties: All polynomials x ∈ K[X] can be decomposed as the product of linear factors Equivalently, there is no proper algebraic extension of K

Section 12 Graphs: Basic

Simple Graph Types: A tuple G = (V,E), V,E are sets Functions: Relations: E Properties: E is composed of unordred pairs of V E is irreﬂexive and symmetric

11 13 GRAPHS: UNDIRECTED GRAPHS

Section 13 Graphs: Undirected Graphs

Undirected graph Types: A graph G = (V,E) Functions: Relations: Properties: ∼ is irreﬂexive and symmetric

Tree Types: A undirectd graph G = (V,E) Functions: Relations: Properties: There are no cycles in G G is connected

Forest Types: An undirected graph G = (V,E) Functions: Relations: Properties: G’s connected component are trees, but G need not be connected

Rooted Tree Types: A tree G Functions: Relations: Properties: One vertex is designated as the root

Line Graph Types: An undirected graph L(G) = (V ∗,E∗), with underlying undirected graph G = (V,E) Functions: f ∶ E → V ∗ Relations: ∼∗= {{x, y} ∈ E∗}S Properties: f maps edges in G to vertexes in L(G) ∗ ∗ ∗ An unordered pair of vertices {vi , vj } ∈ E iﬀ their corresponding edges in G share an endpoint

12 14 GRAPHS: DIRECTED GRAPHS

Section 14 Graphs: Directed Graphs

Simple Directed Graph Types: A graph G = (V,E) Functions: Relations: = {(x, y) ∈ V × V S(x, y) ∈ E} Properties: is irreﬂexive ∀u, v ∈ V, u v ⇏ v u

Polytree Types: A directed graph G = (V,E) Functions: Relations: Properties: G’s underlying undirected graph is a tree

Ordered Tree Types: A directed graph G = (V,E) Functions: Relations: Properties: The children of nodes in G are ordered

Arboresence and Antiarboresence Types: A rooted polytree, G = (V,E) Functions: Relations: Properties: Every node has a directed path away from or toward the root

Cayley Graph Types: A directed graph Γ = Γ(G, S),S a generating set for group G Functions: f ∶ G → V (G) g ∶ S → CS Relations: E(Γ) = {(g, gs) ∈ G × G S ∀g ∈ G, ∀s ∈ S} Properties: f assigns each g ∈ G to a vertex in Γ g assigns each s ∈ S to a unique color The identity element generally is ignored, so the graph does not contain loops

13 15 GRAPHS: K-PARTITE GRAPHS

Graph Algebra Types: Functions: ⋅ ∶ V 2 → V ⋃ {0} Relations: ⎧ ⎪x x, y ∈ V, (x, y) ∈ E Properties: x ⋅ y ∶ ⎨ ⎪ ∈ { } ( ) ∉ ⎩⎪0 x, y V ⋃ 0 , x, y E 0 ∉ V

Section 15 Graphs: K-Partite Graphs

K-partite Graph Types: A graph G = (V,E) Functions: Relations: Properties: V1,V2, ..., Vn are disjoint sets that partition G’s vertices All ei ∈ E can be written as {vi, vj} where vi ∈ Vi, vj ∈ Vj and i ≠ j G is k-colorable

Bipartite Graph Types: A k-partite graph G = (V,E) Functions: Relations: Properties: Special case of k-partite graph for k = 2 V1,V2 are disjoint sets partition G’s vertices All ei ∈ E can be written as {vi, vj} for some vi ∈ V1 and vjinV2 G has no odd cycles G is 2-colorable

Hypercube Graph Types: A regular, bipartite graph Qn = (X,E) Functions: Relations: n n−1 Properties: Qn has 2 vertices and 2 n edges

14 16 GRAPHS: PERFECT GRAPH

Section 16 Graphs: Perfect Graph

Perfect Graph Types: A graph G = (X,E) Functions: ω ∶ G ↦ n χ ∶ G ↦ n Relations: Properties: ω(G) is the the size of the largest clique in G χ(X) is the the chromatic number of G ω(G) = χ(G)

Perfectly Orderable Graphs Types: A perfect graph G = (V,E) Functions: Relations: <= {(x, y) ∈ V × V S x < y} Properties: < is a total order For any chordless path abcd ∈ G and a < b, then d ≮ c

Chordal Graph Types: A perfectly orderable graph G = (V,E) Functions: Relations: Properties: All induced cycles are of order 3 G has a perfect elimination order

Comparability Graph Types: A perfect graph C = (S, ⊥) Functions: Relations: ⊥= {(x, y) ∈ S S x < y} Properties: S is the set of vertices S is a poset under < ⊥ is the edge relation (x, y), (y, z) ∈ E Ô⇒ (x, z) ∈ E

15 17 GRAPHS: NOT FURTHER CATEGORIZED

Section 17 Graphs: Not further categorized

Hyper Graph Types: A tuple H = (V,E) Functions: Relations: Properties: E = {eiSi ∈ Ie, ei ⊆ V } Ie is an index set for edges in H

Connected Graph Types: A graph G = (V,E) Functions: Relations: Properties: There exists a path between any two v ∈ V in the graph

Multigraph Types: Functions: Relations: Properties: E is a multiset More than one edge may connect the same two vertices

Vertex Labeled Graph Types: A graph G = (V,E) Functions: f ∶ V → L Relations: Properties: f maps vertexes to a set of labels L

Edge Labeled Graph Types: A graph G = (V,E) and set L Functions: f ∶ E → L Relations: Properties: f maps edges to a set of labels L

16 18 GROUPS: PRELIMS

Weighted Graph Types: An edge labeled graph G = (V,E) and labels L Functions: Relations: Properties: L is an ordered set

Regular Graph Types: A graph G = (V,E) Functions: Relations: Properties: All v ∈ V are connected to the same number of edges

Section 18 Groups: Prelims

Magma Types: A set M of elements Functions: ⋅ ∶ M 2 → M Relations: Properties: ∀u, v ∈ M ∶ u ⋅ v ∈ M

Semigroup Types: A magma S of elements Functions: Relations: Properties: ⋅ is associative

Monoid Types: A semigroup S Functions: 1 ∈ S Relations: Properties: 1 is the identity for ⋅ 1 ⋅ u = u ⋅ 1 = u

17 19 GROUPS: RANDOM/UNCATAGORIZED

Quasigroup Types: A magma Q Functions: Ó∶ Q2 → Q Ò∶ Q2 → Q 1R, 1L ∈ Q Relations: Properties: Q is a cancellative magma

Loop Types: A quasigroup Q Functions: 1 ∈ Q Relations: Properties: 1 is the identity for ⋅ 1 ⋅ x = x ⋅ 1 = x

Group Types: A set G Functions: ⋅ ∶ G2 → G −−1 ∶ G → G 1 ∈ G Relations: Properties: ⋅ is associative 1 is the identity for ⋅ −−1 is the inverse map for ⋅

Section 19 Groups: Random/Uncatagorized

P Group Types: A group G Functions: Relations: n Properties: ∀g ∈ G, SgS = p , p prime, n ∈ N

18 20 GROUPS: FINITE GROUPS

Torsion Free Group Types: A group G Functions: Relations: n Properties: ∄g ∈ G, g ≠ 1 ∶ g = 1, n ∈ N

Simple Group Types: A group G Functions: Relations: Properties: N ◁ G Ô⇒ N = {1} or G

Topological group Types: A group G Functions: Relations: Properties: −−1 and × are continuous maps

Lie Group Types: A topological group G Functions: Relations: Properties: ∀p ∈ G, ∃Up ≅ Br(p), for some r > 0 −−1 and × are smooth maps

Section 20 Groups: Finite Groups

Finite Group Types: A group G Functions: Relations: Properties: SGS = n, n ∈ N

19 21 GROUPS: FREE GROUPS

Fintie Symmteric Group Types: A ﬁnite group Sn Functions: Relations: Properties: Sn = {σSσ ∶ X → X is bijective }

Permutation Group Types: A group Pn Functions: Relations: Properties: Pn ≤ Sp

Primitive Group Types: A permutation group G that acts on a set X Functions: Relations: Properties: G preserves only trivial partitions

Alternating Group Types: A permutation group An Functions: Relations: Properties: An = {σSσ ∈ Sn ∧ σ even }

Section 21 Groups: Free Groups

Free Group Types: A group F Functions: Relations: Properties: F is generated by a set S There are no relations on FS (beyond group axioms)

20 23 GROUPS: GROUP PRESENTATIONS

Free Abelian Group Types: A group F Functions: Relations: Properties: F is generated by a set S The only relation on F is commutativity

Section 22 Groups: Abelian Groups

Abelian Group Types: A group G Functions: Relations: Properties: ⋅ is commutative

Section 23 Groups: Group Presentations

Finitely Generated Group Types: A group G Functions: Relations: Properties: G is of the form ⟨SSR⟩, S a generating set, R relators SSS = n, n ∈ N

Finitely Presented Group Types: A ﬁnitely generated group G Functions: Relations: Properties: SRS = n, n ∈ N

21 24 GROUPS: CYCLIC GROUPS

Section 24 Groups: Cyclic Groups

Cyclic Group Types: A group G with element a Functions: Relations: Properties: G can be generated from a and a set of relations R G is necessarily commutative

Virtually Cyclic Group Types: A group V with subgroup H Functions: Relations: Properties: ∃H ≤ V ∶ H is cyclic SV ∶ HS = n, n ∈ N

Locally Cyclic Group Types: A group L Functions: Relations: Properties: ∀H ≤ L, if H is ﬁnitely generated, it is cyclic

Cyclically Ordered Group Types: A group C Functions: Relations: [ ,, ] = {(a, b, c) ∈ C × C × CS[a, b, c]} Properties: [a, b, c] is cyclic, asymmetric, transitive and total

Polycyclic Group Types: A group P with subgroups Hi Functions: Relations: Properties: P is necessarily ﬁnitely presented {Hi}i∈{1,2,..,n−1} ∶ Hi~Hi+1 is cyclic

22 26 MODULES: NOT FURTHER CATEGORIZED

Metacyclic Group Types: A group G Functions: Relations: Properties: n ≤ 2

Section 25 Modules: Basic

Module Types: A set M Functions: + ∶ M 2 → M −1 −+ ∶ M → M R × M → R Relations: Properties: + is associative and commutative (M, +) is an abelian group R × M is scalar multiplication by elements in a ring Scalar multiplication is associative and distributes over addition

Section 26 Modules: Not further categorized

Semimodule Types: A set M Functions: + ∶ M 2 → M R × M → M Relations: Properties: + is associative and commutative (M, +) is an commutative monoid R × M is scalar multiplication by elements in a ring Scalar multiplication is associative and distributes over addition

23 27 MODULES: CHAINS

Simple Module Types: A module M Functions: Relations: Properties: The only submodules of M are 0 and M

D Module Types: A module M[X]

Functions: ∂Ri ∶ M → M Relations: Properties: X = {x1, x2, ..., xn} where xi is an indeterminant

∂Ri is in the ring of scalar multiplication

∂Ri satisﬁes the Leibniz product rule

Section 27 Modules: Chains

Noetherian Module Types: A module M Functions: Relations: Properties: For any chain of submodules S0 ⊆ S1 ⊆ ... ⊆ Sk there is some k ≥ 0 s.t. Sk = Sk+1 All submodules of M are ﬁnitely generated

Artinian Module Types: A module M Functions: Relations: Properties: For any chain of submodules S0 ⊇ S1 ⊇ ... ⊇ Sk there is some k ≥ 0 s.t. Sk = Sk+1

24 29 MODULES: TOWARD VECTOR SPACE

Section 28 Modules: Finitely Generated

Finitely Generated Module Types: A module M Functions: Relations: Properties: M has a ﬁnite number of generators

Cyclic Module Types: A ﬁnitely generated module M Functions: Relations: Properties: M is generated from a single element

Section 29 Modules: Toward Vector Space

Torsion Free Module Types: A module M Functions: Relations: Properties: ∄m ∈ M, r, m ≠ 0 ∶ r ⋅ m = 0

Flat Module Types: A torsion Free module M Functions: Relations: Properties: Taking the tensor product over M preserves exact sequences

Projective Module Types: A ﬂat module M Functions: Relations: Properties:

25 30 POSETS AND LATTICES: BASICS

Free Module Types: A projective module M Functions: Relations: Properties: M has no further relations beyond module axioms M has a basis

Fintie DimensionalVector Space Types: A set V over a ﬁeld K Functions: + ∶ V 2 → V −1 −+ ∶ V → V K × V → V 0 ∈ V Relations: Properties: + is associative and commutative K × V is scalar multiplication by K (V, +) is a group

Section 30 Posets and Lattices: Basics

Set Types: a collection of objects Functions: ⊂, ∩, ∪, ∖, ×, ℘, þ Relations: ∈ Properties:

Preordered Set Types: A set P Functions: Relations: ≤= {(x, y) ∈ P × P Sx ≤ y} Properties: All elements need not be comparable under ≤ ≤ transitive and reﬂexive

26 31 POSETS AND LATTICES: POSETS NOT FURTHER CATAGORIZED

Partially Ordered Set Types: A set P Functions: Relations: ≤= {(x, y) ∈ P × P Sx ≤ y} Properties: All elements need not be comparable under ≤ ≤ transitive, reﬂexive and antisymmteric

Section 31 Posets and Lattices: Posets not Further Catagorized

Partially Ordered Space Types: A poset X endowed with topology τ Functions: Relations: ≤= {(x, y) ∈ X × XSx ≤ y} Properties: ∀x, y ∈ X, x ≰ y ∶ ∃U, V ⊂ τ, x ∈ U, y ∈ V ∶ u ≰ v, ∀u ∈ U, ∀v ∈ V

Locally Finite Poset Types: A poset P Functions: Relations: Properties: ∀x, y ∈ P, x ≤ y ∶ [x, y] has ﬁnitely many elements

Partially Ordered group Types: A poset and group G Functions: + ∶ G2 → G −1 −+ ∶ G → G 0 ∈ G Relations: ≤= {(x, y) ∈ G × G S 0 ≤ −x + y} Properties: ≤ respects +

27 33 POSETS AND LATTICES: GRADED STUFF

Section 32 Posets and Lattices: Strict/Total Posets

Strict Poset Types: A poset P Functions: Relations: <= {(x, y) ∈ P 2 S x < y} Properties: < is irreﬂexive, transitive, and asymmetric

Totally Ordered Set Types: A poset P Functions: Relations: Properties: All elements are comparable under ≤

Strict Totally Ordered Set Types: A poset P Functions: Relations: <= {(x, y) ∈ P × P S x < y} Properties: < is irreﬂexive, transitive, asymmetric and trichotomous

Section 33 Posets and Lattices: Graded Stuff

Graded Poset Types: A poset P Functions: ρ ∶ P → N Relations: ⋖= {(x, y) ∈ P S ∄z ∶ x < z < y Properties: x < y Ô⇒ ρ(x) < ρ(y) x ⋖ y Ô⇒ ρ(y) = ρ(x) + 1

Eulerian Poset Types: Functions: Relations: Properties:

28 35 POSETS AND LATTICES: SEMILATTICES

Section 34 Posets and Lattices: Linked + Related

Upward (Downward) Linked Set Types: A subset S of poset P Functions: Relations: Properties: ∀x, y ∈ S ∶ ∃z ∈ P s.t. x ≤ (≥)z and y ≤ (≥)z

Upwards (Downwards) Centered Set Types: A linked subset S of poset P Functions: Relations: Properties: ∀Z ⊆ P,Z has an upper (lower) bound ∈ P

Upward (Downward) Directed Set Types: A preset P Functions: Relations: Properties: ≤ is a preorder ∀x, y ∈ P ∶ ∃z s.t. x ≤ (≥)z and y ≤ (≥)z

Algebraic Poset Types: A poset P Functions: Relations: Properties: each element is the least upper bound of the compact elements below it

Section 35 Posets and Lattices: Semilattices

29 36 POSETS AND LATTICES: LATTICES

Meet Semilatttice: Order Theoretic Edition Types: A poset P Functions: Relations: ≤= {(x, y) ∈ P × P S x ≤ y} Properties: ∀x, y ∈ S, ∃c ∶ c ≤ x, c ≤ y c is the unique greatest lower bound

Meet Semilattice: Algebraic Edition Types: A poset P Functions: ∧ ∶ P 2 → P Relations: Properties: ∧ is associative, commutative, and idempotent

Join Semilatttice: Order Theoretic Edition Types: A poset P Functions: Relations: ≤= {(x, y) ∈ P × P S x ≤ y} Properties: ∀x, y ∈ P, ∃c ∶ x ≤ c, y ≤ c c is the unique least upper bound

Join Semilattice: Algebraic Edition Types: A poset P Functions: ∨ ∶ P 2 → P Relations: Properties: ∨ is associative, commutative, and idempotent

Section 36 Posets and Lattices: Lattices

Lattice (order theory) Types: A set L Functions: Relations: ≤= {(x, y) ∈ L × S x ≤ y} Properties: ∀x, y ∈ L, ∃c, d ∶ c is the greatest lower bound and d is the least upper bound

30 36 POSETS AND LATTICES: LATTICES

Lattice (Algebra) Types: A set L Functions: ∨ ∶ L2 → L ∧ ∶ L2 → L Relations: Properties: (L, ∨), (L, ∧) are both semilattices connected by absorption laws ∀a, b ∈ L ∶ a ∨ (a ∧ b) = a, a ∧ (a ∨ b) = a

Complete Lattice Types: A lattice L Functions: Relations: Properties: ∀A ⊆ L, A has a greatest lower bound and least upper bound

Continuous Lattice Types: A complete lattice A Functions: Relations: ≪= {(x, y) ∈ L × L S ∀D ⊆ L ∶ y ≤ sup D, ∃x ∈ L, ∃d ∈ D s.t. x ≤ d} Properties: ∀x ∈ P, {a S a ≪ x} is directed and has least upper bound x

Algebraic Lattice Types: A continuous lattice A Functions: Relations: Properties: Each element is the least of compact elements below it: those x who satisfy x ≪ a

31 37 POSETS AND LATTICES: A BIG WEB OF LATTICES

Section 37 Posets and Lattices: A Big Web of Lattices

Relatively Complemented Lattice Types: A lattice L Functions: Relations: Properties: ∀c∀d ≥ c, [c, d] ∶ ∀a ∈ [c, d], ∃b s.t.: a ∨ b = d a ∧ b = c a and b are relative complements

Bounded Lattice Types: A lattice L Functions: Relations: Properties: ∀x ∈ L ∶ x ∧ 1 = x x ∨ 1 = 1 x ∨ 0 = x x ∧ 0 = 0

Complemented Lattice Types: A lattice L Functions: Relations: Properties: ∀a ∈ L, ∃b ∈ L ∶ a ∨ b = 1 a ∧ b = 0

Residuated Lattice Types: A lattice L Functions: ⋅ ∶ L2 → L 1 ∈ L Relations: Properties: (L, ⋅, 1) is a monoid (L, ≤) is a lattice ∀z ∶ ∀x, there exists a greatest y and ∀y, there exists a greatest x s.t. x ⋅ y ≤ z

32 37 POSETS AND LATTICES: A BIG WEB OF LATTICES

Atomic Lattice Types: A lattice L Functions: Relations: ⋖= {(x, y) ∈ P S ∄z ∶ x < z < y} Properties: ∀b ∈ L ∶ 0 ∧ b = 0 ∃ai ∈ L ∶ 0 ⋖ ai ∀b ≠ ai, ∃ai ∶ 0 < ai < b {ai}i∈I is the set of atoms in L

Semimodular Lattice Types: A semimodular lattice L Functions: Relations: Properties: a ≤ c Ô⇒ a ∨ (b ∧ c) = (a ∨ b) ∧ c

Modular Lattice Types: A semimodular lattice L Functions: Relations: Properties: a ≤ c Ô⇒ a ∨ (b ∧ c) = (a ∨ b) ∧ c

Distributive Lattice Types: A modular lattice L Functions: Relations: Properties: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)

Orthomodular Lattice Types: A lattice L Functions: Relations: Properties: a ≤ b Ô⇒ b = a ∨ (b ∧ a⊥)

33 37 POSETS AND LATTICES: A BIG WEB OF LATTICES

Orthocomplemented Lattice Types: A lattice L Functions: ⊥∶ L → L Relations: Properties: ⊥ maps an element a to its complement a ∨ a⊥ = 1 a ∧ a⊥ = 0 (a⊥)⊥ = a a ≤ b Ô⇒ a⊥ ≥ b⊥

Geometric Lattice Types: A semimodular, atomic, algebraic Lattice L Functions: Relations: Properties: L is ﬁnite

Heyting Algebra Types: A distributive lattice H Functions: Ð→∶ H2 → H Relations: Properties: (c ∧ a) ≤ b ⇔ c ≤ (a Ð→ b) Monoid (H, ⋅, 1) has operation ⋅ = ∧

Complete Heyting Algebra Types: A heyting algebra H Functions: Relations: Properties: H is complete as a lattice

34 39 RINGS: IDEALS + RELATED

Section 38 Rings: Basics

Ring Types: A set R Functions: ⋅ ∶ R2 → R + ∶ R2 → R −1 −+ ∶ R → R 0 ∈ R Relations: Properties: (R, +) is an abelian group ⋅ is a monoid and distributes over +

Commutative Ring Types: A ring R Functions: Relations: Properties: ⋅ is commutative

Noncommutative Ring Types: A ring R Functions: Relations: Properties: ⋅ need not be commutative

Section 39 Rings: Ideals + Related

Left (right) Ideal Types: A subgroup of a ring, I Functions: Relations: Properties: ∀x ∈ I, ∀r ∈ R ∶ rx (xr) ∈ I

35 39 RINGS: IDEALS + RELATED

Ideal Types: A subgroup of a ring, I Functions: Relations: Properties: The left and right ideals generated by a subgroup are equal ∀x ∈ I, ∀r ∈ R ∶ rx ∈ I

Maximal ideal Types: A subgroup of elements, I Functions: Relations: Properties: I ⊆ J Ô⇒ I = J or J = R

Principal ideal Types: A subgroup of elements, I Functions: Relations: Properties: I is generated by a single element

Prime Ideal Types: A subgroup of elements I Functions: Relations: Properties: ab ∈ I Ô⇒ a ∈ I or b ∈ I

Quotient Ring Types: A ring Q Functions: Relations: ∼= {(x, y} ∈ R × R S x − y ∈ I} Properties: Q is constructed by dividing R with one of its ideals

36 40 RINGS: BOOLEANS + RELATED

Section 40 Rings: Booleans + Related

Boolean Ring Types: A commutative ring R Functions: Relations: Properties: ∀x ∈ R ∶ x2 = x ∀x ∈ R ∶ 2x = 0

Sigma Algebra Types: A subset Σ, of ℘(X), X a set Functions: ∩, ∪, ∖ Relations: Properties: X ∈ Σ ∩, ∪ closed under countable operations

37 41 RINGS: RANDOMS/NOT FURTHER CATAGORIZED

Section 41 Rings: Randoms/Not further catagorized

Semiring Types: A set R Functions: ⋅ ∶ R2 → R +R2 → R 0 ∈ R Relations: Properties: + is a commutative monoid ⋅ is a monoid and distributes over +

Noetherian Ring Types: A ring R Functions: Relations: Properties: For any I0 ⊆ I1 ⊆ ... ⊆ Ik there is some k ≥ 0 s.t. Ik = Ik+1

Artinian Ring Types: A ring R Functions: Relations: Properties: For any I0 ⊇ I1 ⊇ ... ⊇ Ik there is some k ≥ 0 s.t. Ik = Ik+1

Topological Ring Types: A ring R with topology τ Functions: ⋅ ∶ R2 → R ∶ (x, y) ↦ x ⋅ y + ∶ R2 → R ∶ (x, y) ↦ x + y −1 −+ ∶ R → R Relations: Properties: ⋅ distributes over + −1 +, ⋅, −+ are continuous mappings

38 42 RINGS: VALUATION BS

Section 42 Rings: Valuation bs

Domain Types: A ring R Functions: Relations: Properties: ∀a, b ∈ R, ab = 0 Ô⇒ a = 0 or b = 0

Integral Domain Types: A ring R Functions: Relations: Properties: ⋅ is additionally commutative

Unique Factorization Domain Types: An integral domain R Functions: Relations: Properties: All ideals are ﬁnitely generated All irreducible elements are prime ∀x ∈ R, x ≠ 0, x = up1p2...pn for u unit, pi prime

Principal Ideal Domain Types: A unique factorization domain R Functions: Relations: Properties: All ideals are principal Any two elements have a greatet common divisor

Euclidean Domain Types: A principal idea domain R Functions: fi ∶ R ∖ {0} → R Relations: Properties: fi is the euclidean (gcf) algorithim R may have many fi or just one

39 43 RINGS: BUDGET VERSIONS OF FIELDS

Section 43 Rings: Budget versions of fields

Field Types: A ring K Functions: + ∶ K2 → K ×K2 → K −1 −+ ∶ K → K −1 −× ∶ K~{0} → K~{0} 0, 1 ∈ K Relations: Properties: +, × both associate and commute × distributes over +

Module Types: An abelian group (M, +) Functions: ⋅ ∶ r × M → M ⋅ ∶ M × r → M Relations: Properties: ⋅ is scalar multiplication by elements in a ring, R ⋅ associates and distributes over addition

Finite Field Types: A characteristic nonzero ﬁeld K Functions: Relations: Properties: K has a ﬁnite number of elements n The order of some k ∈ K is p for some p prime and n ∈ N

40 45 SETS: COUNTABLE

Section 44 Sets: Basics

Set Types: a collection of objects Functions: ⊂, ∩, ∪, ∖, ×, ℘, SS Relations: ∈ Properties:

Section 45 Sets: Countable

Countable Set Types: A set S Functions: Relations: Properties: SSS ≤ ℵ0

Inﬁnite Countable Set Types: A set S Functions: Relations: Properties: SSS = ℵ0 bijection into N

Finite Countable Set Types: A set S Functions: Relations: Properties: SSS < ℵ0

41 47 SETS: CLASSES

Section 46 Sets: Uncountable

Uncountable Set Types: A set S Functions: Relations: Properties: SSS > ℵ0

Section 47 Sets: Classes

Proper Class Types: a collection, C of sets Functions: ϕ(x) Relations: Properties: ϕ(x) is a membership function C is not a set

Ordinal Types: A collection of sets, Xi Functions: ×, +,Xn,S(n) Relations: > Properties: Xk = {0, 1, 2, ...Xk−1} > is trichotomous, transitive, and wellfounded

Cardinal Types: Ordinals κ Functions: Relations: Properties: The cardinality κ is the least ordinal, α, s.t. there is a bijection between S and α

42 49 TOPOLOGIES: ALGEBRAIC TOPOLOGIES

Section 48 Topologies: Basics

Convergent Space Types: A set S and ﬁlters on S, F Functions: Relations: → Properties: ∀F ∈ F, x ∈ S, F → x ⇔ F convergences to X → is centered, isotone, and directed N (x) = ⋂{F ∈ FSF → x}

Topological Space Types: A set X and τ ⊆ ℘(X) Functions: Relations: Properties: X, ∅ ∈ τ xi ∈ τ Ô⇒ ⋂i∈I xi ∈ τ for ﬁnite I xi ∈ τ Ô⇒ ⋃j∈J xj ∈ τ

Section 49 Topologies: Algebraic Topologies

Topological Group Types: A group G with topology τ Functions: ⋅ ∶ G2 → G ∶ (x, y) ↦ x ⋅ y −−1 ∶ G → G 1 ∈ G Relations: Properties: ⋅, −−1 are continuous mappings

43 49 TOPOLOGIES: ALGEBRAIC TOPOLOGIES

Topological Field Types: A topological ring K Functions: A ﬁeld K with topology τ Relations: ⋅ ∶ K2 → K ∶ (x, y) ↦ x ⋅ y + ∶ K2 → K ∶ (x, y) ↦ x + y −1 −+ ∶ K → R −1 −× ∶ K~{0} → K~{0} Properties:

−1 −1 +, ⋅, −+ , −⋅ are continuous mappings

Topological Vector Space Types: A vector space V over a topological ﬁeld K Functions: + ∶ V 2 → V k × V → V −1 −+ ∶ V → V Relations: Properties: k × V represents scalar multiplication by elements in a ﬁeld K −1 +, −+ and scalar multiplication are continuous

44 50 TOPOLOGIES: WEB I

Section 50 Topologies: Web I

Uniform Space Types: A topological space U and entourages Φ Functions: Relations: U = {(x, y)Sx ≈u y} Properties: Φ is a nonempty collection of relations on S U ∈ Φ and U ⊆ V ⊆ X × X Ô⇒ V ∈ Φ ∀U ∈ Φ, ∃V ∈ Φ, ∀x, y, z ∶ V ○ V ⊆ Φ ∶ x ≈V y, y ≈V z Ô⇒ x ≈U z ∀U ∈ Φ, ∃V ∈ Φ, ∀x, y ∶ V ○ V ⊆ Φ ∶ y ≈V x Ô⇒ x ≈U y U, V ∈ Φ Ô⇒ U ∩ V ∈ Φ

Complete Uniform Space Types: A uniform Space U Functions: Relations: Properties: ∀F ∀U ∶ ∃A ∈ F ∶ A × A ⊆ U, F converges

Manifold Types: A topological space, M Functions: Relations: n Properties: ∀p ∈ M, ∃u ∈ U(p) ∶ u ≅ R Manifolds are not inherently metric or inner product spaces

Compact Manifold Types: A manifold M Functions: Relations: Properties: X = ⋃x∈C x, then for F ⊆ C,F ﬁnite, X = ⋃x∈F x

Smooth Manifold Types: A manifold M Functions: Relations: Properties: Derivatives of arbitrary orders exist

45 51 TOPOLOGIES: WEB II

Pseudo Riemann Manifold Types: A smooth manifold M 2 Functions: gp ∶ TpM → R Relations: Properties: gp is the inner product on the tangent space of a point p gp(X,Y ) = 0 ∀Y Ô⇒ X = 0

Aﬃne Space Ð→ Types: A set A and vector space A Ð→ Functions: A × A → A ∶ (a, v) ↦ a + v Relations: Ð→ Properties: A’s additive group acts freely and transitively on A Ð→ ∀v, w ∈ A, ∀a ∈ A, (a + w) + u = a + (w + u) Ð→ ∀a, b ∈ A, ∃v ∈ R ∶ b = a + v Ð→ ∀a ∈ A, A → A ∶ v ↦ a + v is bijective

Section 51 Topologies: Web II

Metric Space Types: A topological space M 2 Functions: d ∶ M → R Relations: Properties: d(x, y) ≥ 0 d(x, y) = 0 ⇔ x = y d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z)

46 51 TOPOLOGIES: WEB II

Complete Metric Space Types: A metric space M Functions: Relations: Properties: ∀F ∀U ∶ ∃A ∈ F ∶ A × A ⊆ U, F converges

Normed Vector Space Types: A metric and topological vector space V Functions: YY ∶ V → R Relations: Properties: YxY > 0 for x ≠ 0 and YxY = 0 iﬀ x = 0 YαxY = SαSYxY Yx + yY ≤ YxY + YyY YY induces a metric on V ∶ d(x, y) = Yx − yY

Banach Space Types: A vector space V Functions: Relations: Properties: for any Cauchy Sequence {xn}limx→∞Yxn − xY = 0

Inner Product Space Types: A vector space V and ﬁeld of scalars F Functions: ⟨⋅, ⋅⟩ ∶ V 2 → F Relations: Properties: ⟨x, y⟩ = ⟨y, x⟩ ⟨x, x⟩ ≥ 0 ⟨x, x⟩ = 0 iﬀ x = 0 ⟨ax, y⟩ = a⟨x, y⟩ ⟨x + y, z⟩ = ⟨x, z⟩ + ⟨y, z⟩

Hilbert Space Types: An inner product space H Functions: Relations: » Properties: A norm may be deﬁned as YxY = ⟨x, x⟩ for any Cauchy Squence {xn}, limx→∞Yxn − xY = 0

47 51 TOPOLOGIES: WEB II

Euclidean Space n Types: A banach and hilbert space R Functions: Relations: » » n 2 Properties: A norm is» deﬁned as YxY = ⟨x, x⟩ = ∑i=1(xi) n 2 d(x, y) = ∑i=1(xi − yi)