The Category of Automatic Groups
The Construction of an Automatic Group
Creation of Automatic Groups and Arithmetic with Words
Construction of a Automatic Group
AutomaticGroup(Q: parameters) : GrpFP -> GrpAtc
SetVerbose("KBMAG", v) : MonStgElt, RngIntElt ->
Example GrpAtc_AutomaticGroup (H31E1)
Construction of a Word
Identity(G) : GrpAtc -> GrpAtcElt
G ! [ i_1, ..., i_s ] : GrpAtc, [ RngIntElt ] -> GrpAtcElt
Example GrpAtc_Words (H31E2)
Arithmetic with Words
u * v : GrpAtcElt, GrpAtcElt -> GrpAtcElt
u / v : GrpAtcElt, GrpAtcElt -> GrpAtcElt
u ^ n : GrpAtcElt, RngIntElt -> GrpAtcElt
u ^ v : GrpAtcElt, GrpAtcElt -> GrpAtcElt
Inverse(w) : GrpAtcElt -> GrpAtcElt
(u, v) : GrpAtcElt, GrpAtcElt -> GrpAtcElt
(u_1, ..., u_r) : GrpAtcElt, ..., GrpAtcElt -> GrpAtcElt
u eq v : GrpAtcElt, GrpAtcElt -> BoolElt
u ne v : GrpAtcElt, GrpAtcElt -> BoolElt
IsId(w) : GrpAtcElt -> BoolElt
# u : GrpAtcElt -> RngIntElt
ElementToSequence(u) : GrpAtcElt -> [ RngIntElt ]
Example GrpAtc_Arithmetic (H31E3)
Accessing Group Information
G . i : GrpAtc, RngIntElt -> GrpAtcElt
Generators(G) : GrpAtc -> [GrpAtcElt]
NumberOfGenerators(G) : GrpAtc -> RngIntElt
Relations(G) : GrpAtc -> [GrpFPRel]
NumberOfRelations(G) : GrpAtc -> RngIntElt
Ordering(G) : GrpAtc -> String
Parent(w) : GrpAtcElt -> GrpAtc
Example GrpAtc_BasicAccess (H31E4)
Construction of Homomorphisms
hom< A -> G | S > : Struct , Struct -> Map
Operations on the Set of Elements
Order Functions
Order(G) : GrpAtc -> RngIntElt
IsFinite(G) : GrpAtc -> BoolElt, RngIntElt
Example GrpAtc_Order (H31E5)
Set Operations
Random(G, n) : GrpAtc, RngIntElt -> GrpAtcElt
Random(G) : GrpAtc -> GrpAtcElt
Representative(G) : GrpAtc -> GrpAtcElt
Set(G, a, b) : GrpAtc, RngIntElt, RngIntElt -> SetEnum
Set(G) : GrpAtc -> SetEnum
Seq(G, a, b) : GrpAtc, RngIntElt, RngIntElt -> SeqEnum
Seq(G) : GrpAtc -> SeqEnum
Example GrpAtc_Set (H31E6)
Membership and Equality
w in G : GrpAtcElt, GrpAtc -> BoolElt
w notin G : GrpAtcElt, GrpAtc -> BoolElt
S subset G : { GrpAtcElt }, GrpAtc -> BoolElt
S notsubset G : { GrpAtcElt }, GrpAtc -> BoolElt
Properties of a Automatic Group
Automatic Group Predicates
IsConfluent(G) : GrpAtc -> BoolElt
Example GrpAtc_IsConfluent (H31E7)
Accessing Automata
GrowthFunction(G) : GrpAtc -> FldFunRatElt
Example GrpAtc_GrowthFunction (H31E8)