Theoretical Background
On the following pages you will find an overview of mathematical concepts which are useful to know when working with Algebra WorkBench.
You can either read through the pages in order - or you can look up specific concepts with our alphabetical index.
If you're looking for more detailed information, there is a pre-print version of the lecture notes (only in English) for the course "Mathematische Logik" by Prof. Jürg Schmid available for download. Click here to download these notes in pdf-format.
Definitions and Propositions [top]
- Order Theory
- Universal Algebra
- Additional Background
Alphabetical Index [top]
Symbols, etc A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Symbols, etc
- 0,1-simple
- 0,1-simple, Satz
- 0,1-simpler Verband (Vermutung)
- 0-1-trennend
- 0-1 sublattice
- 0-1-Vermutung
- σ(f)-stellig
- ΓUL
- ↑ (up-set)
- ↓ (down-set)
- ∧-dense
- ∧-irreducible (meet-irreducible)
- ∨-dense
- ∨-irreducible (join-irreducible)
A
- AL
- AU
- Algebra (cf. also macroproblems 6, 9)
- Algebra, boolesche (siehe auch: AWB Enstiegsprobleme 5, 6, 8)
- Algebra, boolesche (in Beweis Komplementbehauptung)
- Algebra, einfache
- algebraizable
- Allrelation
- antichain
- Äquivalenklasse
- Äquivalenzrelation
- Äquivalenzrelationenverband (cf. also introductory problem 4)
- Automorphism
B
- Behauptung, Drei-Stränge
- Bild, homomorphes
- Birkhoff, Satz von
- boolesche Algebra (siehe auch: AWB Einstiegsprobleme 5, 6, 8)
- bound
C
- chain (cf. also macroproblems 5, 7)
- closure-operator
- comparable
- completeness
- cover
- critical pair (cf. also macroproblem 7)
D
- Dedekind-MacNeille
- Dedekind-MacNeille-Vervollständigung
- Diagonale
- dense
- direktes Produkt (siehe auch: AWB Produkteregister, Einstiegsprobleme 5,6, macro problem 8)
- distributiver Verband (cf. also macroproblem 7)
- down-set
- Drei-Stränge Behauptung
- dual poset
E
- Einbettung
- einfache Algebra
- embedding, order-
- endomorphism (cf. also macroproblem 6)
- Epimorphism
- extreme elements
F
- fast rigid (cf. also macroproblem 8)
- fundamentale Operation
G
H
- Halbverband (cf. also macroproblem 8)
- Hashimoto Proposition
- Hasse diagram
- homomorphes Bild
- homomorphism (cf. also: AWB introductory problem 5, macro problem 8)
- homomorphism, order-
I
- incomparable
- infimum
- Intervall (cf. also macroproblem 9)
- irreducible element
- Isomorphism (siehe auch AWB Einstiegsproblem 9)
- isomorphism, order-
J
K
- Kern
- Komplement
- Komplement (in Beweis Komplementbehauptung)
- Komplementbehauptung
- Kongruenz, Kongruenzrelation (siehe auch: AWB Kongruenzen-Register, Einstiegsprobleme 4, 9, macroproblem 9)
- kongruenzdistributiv (Definition in Einstiegsproblem 6)
- Kongruenzklasse
- Kongruenzverband (siehe auch: AWB Einstiegsproblem 6)
L
- lattice (definition 1) (cf. also macroproblem 9)
- lattice, modular
- least upper bound
- lower bound
- lower cover (neighbor)
M
- maximal 0-1 sublattice (cf. also macroproblem 7)
- maximal element
- maximum
- meet
- minimal element
- minimum
- modular lattice
- Morphismus (siehe auch: AWB Morphismenfenster,Makroproblem 3)
N
O
- Operation (cf. also introductory problem 3)
- Operationssymbole
- order-embedding
- order-homomorphism
- order-isomorphism
P
- partial order (cf. also macroproblem 4)
- partially ordered set
- PCS (pseudokomplementierter Halbverband, p-semilattice)
- poset
- poset, dual
- Produkt, direktes (siehe auch: AWB Produkteregister, Einstiegsprobleme 5, 6, 8)
- Produkt, subdirektes (cf. also macroproblem 9)
- proper weakly critical (subcritical) (cf. also macroproblem 7)
- Pseudokomplement, Schnitt-, Verein (meet-, join-) (cf. also introductory problem 5)
- Pseudokomplement (in Beweis Komplementbehauptung)
Q
R
S
- Satz 0,1-simple
- Satz Drei-Stränge Behauptung
- Satz von Birkhoff
- Schnitt-Halbverband
- Schnitt-Pseudokomplement (meet-pseudocomplement)
- subcritical (weakly critical) pair (cf. also macroproblem 7)
- subdirektes Produkt (cf. also macroproblem 9)
- subposet
- supremum
- Skelett (cf. also macroproblem 8)
- stark extensiv (Definition in macroproblem 6)
- Strang
T
- Translation (cf. also macroproblem 5)
- Typ
U
- Unteralgebra (siehe auch: AWB Substrukturen-Register, Einstiegsprobleme 8, 9, 10, Makroproblem 7)
- Unteralgebrenverband (siehe auch AWB Einstiegsproblem 10)
- up-set
- upper bound
- upper cover (neighbor)
V
- Verband (Definition 2)
- Verband, 0,1-simpler (Vermutung)
- Verband, distributiver (cf. also macroproblem 7)
- Verein-Pseudokomplement (join-pseudocomplement)
- Vermutung über 0,1-simple Verbände
- verträglich
W
- weakly critical (subcritical) pair (cf. also macroproblem 7)