diff --git a/lib/lib_local.gd b/lib/lib_local.gd index a4e15a6..5ddf891 100644 --- a/lib/lib_local.gd +++ b/lib/lib_local.gd @@ -79,9 +79,9 @@ ################################### #! @Description -#! The argument is $n$. -#! The output a list of IdGroup of the additive groups -#! of local nearrings from Library of order $n$. +#! The argument is $n$. +#! The output is a list of additive groups of local nearrings in the +#! library of this package of order $n$. #! @Returns a list #! @Arguments n #! @Label @@ -96,9 +96,9 @@ DeclareGlobalFunction( "AdditiveGroupsOfLibraryOfLNRsOfOrder"); ################################### #! @Description -#! The argument is a group $G$. -#! The output a list of the catalogues of local nearrings -#! from Library on $G$. +#! The argument is a group $G$. +#! The output is a list of catalogue entries for the local nearrings in +#! the library of this package whose additive group is isomorphic to $G$. #! @Returns a list #! @Arguments G #! @Label @@ -122,9 +122,10 @@ DeclareGlobalFunction( "LibraryOfLNRsOnGroup"); #! @Description #! The arguments are $k$, $l$, $m$, $n$, $w$. -#! The output is local nearring from Library without -#! check. The arguments $k$, $l$, $m$, $n$ are from IdGroup of the additive group and the multiplicative group, -#! respectively, $w$ is the position in the list. +#! The output is the $w$-th local nearring from the library of this +#! package whose additive group has IdGroup value [k,l] +#! and whose multiplicative group has IdGroup value [m,n]. +#! No validation of the arguments is performed. #! @Returns a nearring #! @Arguments k,l,m,n,w #! @Label @@ -140,8 +141,10 @@ DeclareOperation( "LocalNearRing", [ IsInt, IsInt, IsInt, IsInt, IsInt ]); #! @Description #! The arguments are $k$, $l$, $m$, $n$. -#! The output are all local nearrings from Library without -#! check. The arguments $k$, $l$, $m$, $n$ are as above. +#! The output is the list of all local nearrings from the library of this +#! package whose additive group has IdGroup value [k,l] +#! and whose multiplicative group has IdGroup value [m,n]. +#! No validation of the arguments is performed. #! @Returns a list #! @Arguments k,l,m,n #! @Label @@ -157,8 +160,10 @@ DeclareOperation( "AllLocalNearRings", [ IsInt, IsInt, IsInt, IsInt ]); #! @Description #! The arguments are $k$, $l$, $m$, $n$. -#! The output are number of all local nearrings from Library without -#! check. The arguments $k$, $l$, $m$, $n$ are as above. +#! The output is the number of local nearrings in the library of this +#! package whose additive group has IdGroup value [k,l] +#! and whose multiplicative group has IdGroup value [m,n]. +#! No validation of the arguments is performed. #! @Returns a number #! @Arguments k,l,m,n #! @Label @@ -174,10 +179,10 @@ DeclareSynonym("NrLocalNearRings", NumberLocalNearRings); ################################### #! @Description -#! The argument is a group $G$. -#! The output is true if in Library there exists a local nearring -#! whose additive group is isomorphic to $G$ -#! otherwise the output is false. +#! The argument is a group $G$. +#! The output is true if the library of this package contains a +#! local nearring whose additive group is isomorphic to $G$, and +#! false otherwise. #! @Returns a boolean #! @Arguments G #! @Label @@ -195,9 +200,9 @@ DeclareGlobalFunction( "IsAdditiveGroupOfLibraryOfLNRs"); ################################### #! @Description -#! The argument is a group $G$. -#! The output some information about local nearrings -#! from Library on $G$. +#! The argument is a group $G$. +#! The output is summary information about the local nearrings in the +#! library of this package whose additive group is isomorphic to $G$. #! @Returns information #! @Arguments G #! @Label diff --git a/lib/local.gd b/lib/local.gd index 76d68a5..81076d0 100644 --- a/lib/local.gd +++ b/lib/local.gd @@ -30,8 +30,8 @@ #! @Label DeclareProperty( "IsMinimalNonAbelianGroup", IsGroup ); -#! Recall that each finite non-abelian group whose proper subgroups are -#! abelian is called a Miller-Moreno group or in other terminology +#! Recall that each finite non-abelian group whose proper subgroups are +#! abelian is called a Miller-Moreno group or, in other terminology, #! a minimal non-abelian group. #! @BeginExample @@ -73,7 +73,9 @@ DeclareProperty( "IsMetacyclicPGroup", IsPGroup ); #! @Description #! The argument is a group $G$. -#! The output is +#! The output is a list of pairs [x,H], where x is a +#! representative of an endo-orbit of $G$ and H is the set of all +#! images of x under endomorphisms of $G$. #! @Returns EndoOrbitsOfGroup #! @Arguments G #! @Label @@ -93,19 +95,19 @@ DeclareOperation( "EndoOrbitsOfGroup", [ IsGroup ] ); #! @Description #! The argument is a group $G$. -#! The output is true if $G$ is a endocyclic group, +#! The output is true if $G$ is an endocyclic group, #! otherwise the output is false. #! @Returns a boolean #! @Arguments G #! @Label DeclareProperty( "IsEndoCyclicGroup", IsGroup ); -#!Let $G$ be a group and $End G$ be the set of all its endomorphisms, -#!which can be considered as a semigroup with respect to the composition operation of endomorphisms. -#!For each $g\in G$ we denote by $g^{End G}$ the set $\{g^\alpha| \alpha\in End G\}$ +#! Let $G$ be a group and $End G$ be the set of all its endomorphisms, +#! which can be considered as a semigroup with respect to composition. +#! For each $g\in G$ we denote by $g^{End G}$ the set $\{g^\alpha| \alpha\in End G\}$ #!of all images of the element $g$ with respect to endomorphisms of $End G$. -#!A group $G$ is called endocyclic if it contains an element $g$ with $G=g^{End G}$. +#! A group $G$ is called endocyclic if it contains an element $g$ with $G=g^{End G}$. #! @BeginExample #! gap> IsEndoCyclicGroup(D); @@ -347,16 +349,17 @@ DeclareProperty( "IsOneGeneratedNearRing", IsNearRing ); #! @Description #! The arguments are a nearring $R$ with identity and a set of units $Un$ of $R$. -#! The output are the automorphisms associated with nearring units. -#! @Returns automorphisms +#! The output is the list of automorphisms of the additive group of $R$ +#! associated with the units in $Un$. +#! @Returns a list of automorphisms #! @Arguments R,Un #! @Label DeclareOperation( "AutomorphismsAssociatedWithNearRingUnits", [ IsNearRing, IsNearRingElementCollection ] ); #! A subgroup $A$ of the automorphism group $Aut R^+$ of the additive group of -#! the nearring $R$ with identity isomorphic to the multiplicative group $R^*$ -#! and satisfies the condition $$i^A=\{i^a\mid a\in A\}=R^*$$ is called -#! the subgroup of $Aut R^+$ associated with the group $R^*$. +#! the nearring $R$ with identity isomorphic to the multiplicative group $R^*$ +#! and satisfying the condition $$i^A=\{i^a\mid a\in A\}=R^*$$ is called +#! the subgroup of $Aut R^+$ associated with $R^*$. #! @BeginExample @@ -377,7 +380,7 @@ DeclareOperation( "AutomorphismsAssociatedWithNearRingUnits", [ IsNearRing, IsNe #! @Description #! The arguments are a nearring $R$ and a set $Elm$ of nearring elements. #! The output is the endomorphisms associated with nearring elements. -#! @Returns endomorphisms +#! @Returns a list of endomorphisms #! @Arguments R, Elm #! @Label DeclareOperation( "EndomorphismsAssociatedWithNearRingElements", [ IsNearRing, IsNearRingElementCollection ] ); @@ -394,7 +397,7 @@ DeclareOperation( "EndomorphismsAssociatedWithNearRingElements", [ IsNearRing, I #! @Description #! The argument is a nearring $R$ with identity. -#! The output is the semidirect product associated with nearring $R$. +#! The output is the semidirect product associated with the nearring $R$. #! @Returns a semidirect product #! @Arguments R #! @Label @@ -413,8 +416,8 @@ DeclareOperation( "SemidirectProductAssociatedWithNearRing", [ IsNearRing ]); #! @Description #! The arguments are a nearring $R$ with identity and -#! a subgroup $H$ of additive group of $R$. -#! The output is true if $H$ is a constructive subgroup of nearring $R$, +#! a subgroup $H$ of the additive group of $R$. +#! The output is true if $H$ is a circle subgroup of the nearring $R$, #! otherwise the output is false. #! @Returns a boolean #! @Arguments R,H @@ -433,8 +436,9 @@ DeclareOperation( "IsCircleSubgroupOfNearRing", [IsNearRing, IsGroup ]); #! @Description #! The arguments are a nearring $R$ with identity and -#! a constructive subgroup $H$ of $R$. -#! The output is the group +#! a circle subgroup $H$ of $R$. +#! The output is the semidirect product associated with $H$ and the +#! automorphisms determined by the corresponding units. #! @Returns a group #! @Arguments R,H #! @Label @@ -451,7 +455,7 @@ DeclareOperation( "FactorizedGroupAssociatedWithCircleSubgroupOfNearRing", [ IsN #! @Description #! The argument is a nearring $R$. -#! The output is the constant part of nearring $R$. +#! The output is the constant part of the nearring $R$. #! @Returns a constant part #! @Arguments R #! @Label @@ -470,7 +474,7 @@ DeclareAttribute("ConstantPartOfNearRing", IsNearRing ); #! @Description #! The argument is a nearring $R$. -#! The output is the zero-symmetric part of nearring $R$. +#! The output is the zero-symmetric part of the nearring $R$. #! @Returns a zero-symmetric part #! @Arguments R #! @Label @@ -486,8 +490,9 @@ DeclareAttribute("ZeroSymmetricPartOfNearRing", IsNearRing ); #! @Description #! The argument is a nearring $R$. -#! The output is the group of units as group of automorphisms $R$. -#! @Returns a group of units +#! The output is a group of automorphisms of the additive group of $R$ +#! that is isomorphic to the group of units of $R$. +#! @Returns a group #! @Arguments R #! @Label DeclareAttribute("GroupOfUnitsAsGroupOfAutomorphisms", IsNearRing ); @@ -505,7 +510,7 @@ DeclareAttribute("GroupOfUnitsAsGroupOfAutomorphisms", IsNearRing ); ################################### #! @Description #! The argument is a nearring $R$ and an element $r$. -#! The output is true if $r$ is a distributive element of nearring $R$, +#! The output is true if $r$ is a distributive element of the nearring $R$, #! otherwise the output is false. #! @Returns a boolean #! @Arguments R, r diff --git a/tst/localnr01.tst b/tst/localnr01.tst index cfe5435..d9f3f13 100644 --- a/tst/localnr01.tst +++ b/tst/localnr01.tst @@ -10,12 +10,12 @@ # gap> START_TEST("localnr01.tst"); -# doc/_Chapter_Local_nearrings.xml:76-80 +# doc/_Chapter_Local_nearrings.xml:77-81 gap> List(AdditiveGroupsOfLibraryOfLNRsOfOrder(81),IdGroup); [ [ 81, 1 ], [ 81, 2 ], [ 81, 3 ], [ 81, 5 ], [ 81, 6 ], [ 81, 11 ], [ 81, 12 ], [ 81, 13 ], [ 81, 15 ] ] -# doc/_Chapter_Local_nearrings.xml:96-105 +# doc/_Chapter_Local_nearrings.xml:95-104 gap> G:=SmallGroup(81,2); gap> LibraryOfLNRsOnGroup(G); @@ -25,21 +25,21 @@ gap> LibraryOfLNRsOnGroup(G); "AllLocalNearRings(81,2,72,14)", "AllLocalNearRings(81,2,72,19)", "AllLocalNearRings(81,2,72,24)", "AllLocalNearRings(81,2,72,26)" ] -# doc/_Chapter_Local_nearrings.xml:121-125 +# doc/_Chapter_Local_nearrings.xml:119-123 gap> L:=LocalNearRing(81,12,54,8,3); ExplicitMultiplicationNearRing ( , multiplication ) -# doc/_Chapter_Local_nearrings.xml:140-144 +# doc/_Chapter_Local_nearrings.xml:138-142 gap> L:=AllLocalNearRings(81,12,54,8);; gap> Size(L); 30 -# doc/_Chapter_Local_nearrings.xml:159-162 +# doc/_Chapter_Local_nearrings.xml:157-160 gap> NumberLocalNearRings(81,15,54,8); 10 -# doc/_Chapter_Local_nearrings.xml:178-185 +# doc/_Chapter_Local_nearrings.xml:174-181 gap> G:=SmallGroup(25,2); gap> IsAdditiveGroupOfLibraryOfLNRs(G); @@ -47,7 +47,7 @@ true gap> IsAdditiveGroupOfLibraryOfLNRs(SmallGroup(81,14)); false -# doc/_Chapter_Local_nearrings.xml:200-209 +# doc/_Chapter_Local_nearrings.xml:194-203 gap> InfoLocalNearRing(SmallGroup(361,2)); The local nearrings are sorted by their multiplicative groups. [ "AllLocalNearRings(361,2,342,1) (2)", "AllLocalNearRings(361,2,342,2) (2)", \ diff --git a/tst/localnr02.tst b/tst/localnr02.tst index b75840d..7ac174c 100644 --- a/tst/localnr02.tst +++ b/tst/localnr02.tst @@ -10,7 +10,7 @@ # gap> START_TEST("localnr02.tst"); -# doc/_Chapter_Functions.xml:25-36 +# doc/_Chapter_Functions.xml:24-35 gap> H:=SmallGroup(120,4); gap> IsMinimalNonAbelianGroup(H); @@ -22,7 +22,7 @@ true gap> IsMinimalNonAbelianGroup(SmallGroup(16,8)); false -# doc/_Chapter_Functions.xml:51-58 +# doc/_Chapter_Functions.xml:48-55 gap> IsMetacyclicPGroup(K); true gap> IsMetacyclicPGroup(SmallGroup(81,4)); @@ -30,7 +30,7 @@ true gap> IsMetacyclicPGroup(SmallGroup(81,15)); false -# doc/_Chapter_Functions.xml:72-80 +# doc/_Chapter_Functions.xml:69-77 gap> D:=SmallGroup(81,2); gap> T:=EndoOrbitsOfGroup(D);; @@ -39,11 +39,11 @@ gap> Length(T); gap> Size(T[1][2]); 81 -# doc/_Chapter_Functions.xml:100-103 +# doc/_Chapter_Functions.xml:95-98 gap> IsEndoCyclicGroup(D); true -# doc/_Chapter_Functions.xml:124-140 +# doc/_Chapter_Functions.xml:116-132 gap> N:=LocalNearRing(32,5,16,3,8); ExplicitMultiplicationNearRing ( , multiplication ) @@ -60,7 +60,7 @@ LibraryNearRing(6/2, 3) gap> UnitsOfNearRing(L); [ ] -# doc/_Chapter_Functions.xml:155-169 +# doc/_Chapter_Functions.xml:145-159 gap> H:=SmallGroup(16,6); gap> A:= AutomorphismNearRing(H); @@ -75,7 +75,7 @@ LibraryNearRing(8/2, 814) gap> IsLocalNearRing(K); false -# doc/_Chapter_Functions.xml:184-191 +# doc/_Chapter_Functions.xml:172-179 gap> L:=AllLocalNearRings(16,14,8,4);; gap> Size(L); 24 @@ -83,7 +83,7 @@ gap> F:=Filtered(L,x->IsLocalRing(x));; gap> Size(F); 1 -# doc/_Chapter_Functions.xml:205-220 +# doc/_Chapter_Functions.xml:191-206 gap> T:=LocalNearRing(49,2,42,1,1); ExplicitMultiplicationNearRing ( , multiplication ) @@ -99,7 +99,7 @@ gap> N:=SortedList(NearRingNonUnits(R)); ((1,5,3,7)(2,8,4,6)), ((1,6,3,8)(2,5,4,7)), ((1,7,3,5)(2,6,4,8)), ((1,8,3,6)(2,7,4,5)) ] -# doc/_Chapter_Functions.xml:234-247 +# doc/_Chapter_Functions.xml:218-231 gap> B:=LocalNearRing(25,2,20,3,1); ExplicitMultiplicationNearRing ( , multiplication ) gap> D:=DistributiveElements(B);; @@ -113,7 +113,7 @@ false gap> IsDgNearRing(Rs); true -# doc/_Chapter_Functions.xml:261-271 +# doc/_Chapter_Functions.xml:243-253 gap> T:=LocalNearRing(125,4,100,9,1); ExplicitMultiplicationNearRing ( , multiplication ) @@ -124,26 +124,26 @@ Group([ of ..., f2, f3, f2^2, f2*f3, f3^2, f2^3, f2^2*f3, f2*f3^2, f3 gap> IdGroup(L); [ 25, 2 ] -# doc/_Chapter_Functions.xml:285-290 +# doc/_Chapter_Functions.xml:265-270 gap> I:=NonUnitsAsNearRingIdeal(T); < nearring ideal > gap> Size(I); 25 -# doc/_Chapter_Functions.xml:304-310 +# doc/_Chapter_Functions.xml:282-288 gap> B:=LocalNearRing(16,10,8,2,7);; gap> M:=MultiplicativeSemigroupOfNearRing(B); gap> Size(M); 16 -# doc/_Chapter_Functions.xml:324-329 +# doc/_Chapter_Functions.xml:300-305 gap> Nm:=NonUnitsAsMultiplicativeSemigroup(B); gap> Size(Nm); 8 -# doc/_Chapter_Functions.xml:344-354 +# doc/_Chapter_Functions.xml:318-328 gap> D:=LocalNearRing(49,2,42,4,1); ExplicitMultiplicationNearRing ( , multiplication ) gap> IsOneGeneratedNearRing(D); @@ -154,7 +154,7 @@ ExplicitMultiplicationNearRing ( IsOneGeneratedNearRing(H); false -# doc/_Chapter_Functions.xml:372-383 +# doc/_Chapter_Functions.xml:345-356 gap> S:=UnitsOfNearRing(D); [ (f1), (f1*f2), (f1*f2^2), (f1*f2^3), (f1*f2^4), (f1*f2^5), (f1*f2^6), (f1^2), (f1^2*f2), (f1^2*f2^2), (f1^2*f2^3), (f1^2*f2^4), (f1^2*f2^5), (f1^2*f2^6), (f1^3), (f1^3*f2), @@ -166,14 +166,14 @@ gap> A:=AutomorphismsAssociatedWithNearRingUnits(D,S);; gap> Size(A); 42 -# doc/_Chapter_Functions.xml:397-403 +# doc/_Chapter_Functions.xml:368-374 gap> Nu:=NearRingNonUnits(D); [ ( of ...), (f2), (f2^2), (f2^3), (f2^4), (f2^5), (f2^6) ] gap> En:=EndomorphismsAssociatedWithNearRingElements(D,Nu);; gap> Size(En); 7 -# doc/_Chapter_Functions.xml:417-424 +# doc/_Chapter_Functions.xml:386-393 gap> T:=LocalNearRing(25,2,20,2,1); ExplicitMultiplicationNearRing ( , multiplication ) gap> SemidirectProductAssociatedWithNearRing(T); @@ -181,20 +181,20 @@ gap> SemidirectProductAssociatedWithNearRing(T); gap> Size(last); 500 -# doc/_Chapter_Functions.xml:440-446 +# doc/_Chapter_Functions.xml:407-413 gap> Sg:=Subgroups(GroupReduct(T));; gap> Size(Sg); 8 gap> F:=Filtered(Sg,x->IsCircleSubgroupOfNearRing(T,x)); [ Group([ ]), Group([ f2 ]) ] -# doc/_Chapter_Functions.xml:461-466 +# doc/_Chapter_Functions.xml:427-432 gap> FG:=FactorizedGroupAssociatedWithCircleSubgroupOfNearRing(T,F[2]); gap> IdGroup(FG); [ 25, 2 ] -# doc/_Chapter_Functions.xml:480-487 +# doc/_Chapter_Functions.xml:444-451 gap> H:=LocalNearRing(361,2,342,7,7); ExplicitMultiplicationNearRing ( , multiplication ) @@ -202,12 +202,12 @@ gap> C:=ConstantPartOfNearRing(H);; gap> Size(C); 19 -# doc/_Chapter_Functions.xml:501-505 +# doc/_Chapter_Functions.xml:463-467 gap> ZeroSymmetricPartOfNearRing(H);; gap> Size(last); 19 -# doc/_Chapter_Functions.xml:519-526 +# doc/_Chapter_Functions.xml:480-487 gap> M:=LocalNearRing(27,4,18,3,2); ExplicitMultiplicationNearRing ( , multiplication ) gap> GroupOfUnitsAsGroupOfAutomorphisms(M); @@ -215,7 +215,7 @@ gap> GroupOfUnitsAsGroupOfAutomorphisms(M); gap> Size(last); 18 -# doc/_Chapter_Functions.xml:541-550 +# doc/_Chapter_Functions.xml:500-509 gap> D:=LocalNearRing(49,2,42,6,1); ExplicitMultiplicationNearRing ( , multiplication ) @@ -225,13 +225,13 @@ gap> d:=h[3]; gap> IsDistributiveElementOfNearRing(D,d); true -# doc/_Chapter_Functions.xml:565-570 +# doc/_Chapter_Functions.xml:522-527 gap> N:=LocalNearRing(16,10,8,2,7); ExplicitMultiplicationNearRing ( , multiplication ) gap> IsSemiDistributiveNearRing(N); true -# doc/_Chapter_Functions.xml:585-595 +# doc/_Chapter_Functions.xml:540-550 gap> N:=LocalNearRing(343,5,294,8,2); ExplicitMultiplicationNearRing ( , multiplication ) @@ -242,7 +242,7 @@ gap> Identity(N); gap> IsNearRingWithIdentity(N); true -# doc/_Chapter_Functions.xml:611-627 +# doc/_Chapter_Functions.xml:564-580 gap> T:=LocalNearRing(49,2,42,1,2); ExplicitMultiplicationNearRing ( , multiplication )