  
  [1X85 [33X[0;0YFunction-Operation-Attribute Triples[133X[101X
  
  [33X[0;0Y[5XGAP[105X  is  eager to maintain information that it has gathered about an object,
  possibly   by   lengthy  calculations.  The  most  important  mechanism  for
  information  maintenance  is  the  automatic  storage and look-up that takes
  place  for  [13Xattributes[113X; and this was already mentioned in section [14X'Tutorial:
  Attributes'[114X.  In this chapter we will describe further mechanisms that allow
  storage of results that are not values of attributes.[133X
  
  [33X[0;0YThe  idea  which is common to all sections is that certain operations, which
  are  not  themselves  attributes, have an attribute associated with them. To
  automatically delegate tasks to the attribute, [5XGAP[105X knows, in addition to the
  [13Xoperation[113X  and  the  [13Xattributes[113X also a [13Xfunction[113X, which is [21Xwrapped around[121X the
  other  two.  This [21Xwrapper function[121X is called by the user and decides whether
  to  call  the  operation  or  the  attribute  or  possibly  both.  The whole
  [13Xf[113Xunction-[13Xo[113Xperation-[13Xa[113Xttribute  triple  (or  [13XFOA triple[113X) is set up by a single
  [5XGAP[105X  command  which  writes  the  wrapper function and already installs some
  methods,  e.g., for the attribute to fall back on the operation. The idea is
  then  that  subsequent  methods,  which  perform the actual computation, are
  installed  only  for  the  operation,  whereas  the wrapper function remains
  unaltered,  and  in  general  no  additional  methods  for the attribute are
  required either.[133X
  
  
  [1X85.1 [33X[0;0YKey Dependent Operations[133X[101X
  
  [1X85.1-1 KeyDependentOperation[101X
  
  [33X[1;0Y[29X[2XKeyDependentOperation[102X( [3Xname[103X, [3Xdom-req[103X, [3Xkey-req[103X, [3Xkey-test[103X ) [32X function[133X
  
  [33X[0;0YThere are several functions that require as first argument a domain, e.g., a
  group,  and  as  second  argument  something  much  simpler,  e.g., a prime.
  [2XSylowSubgroup[102X  ([14X39.13-1[114X)  is  an  example.  Since  its  value depends on two
  arguments,  it cannot be an attribute, yet one would like to store the Sylow
  subgroups once they have been computed.[133X
  
  [33X[0;0YThe   idea   is   to   provide   an   attribute   of   the   group,   called
  [10XComputedSylowSubgroups[110X,  and  to  store  the  groups  in this list. The name
  implies  that  the value of this attribute may change in the course of a [5XGAP[105X
  session,  whenever  a  newly-computed  Sylow  subgroup is put into the list.
  Therefore,  this is a [13Xmutable attribute[113X (see [14X13.5[114X). The list contains primes
  in  each  bound  odd  position  and  a  corresponding  Sylow subgroup in the
  following even position. More precisely, if [10X[3Xp[103X[10X = ComputedSylowSubgroups( [3XG[103X[10X )[
  [3Xeven[103X[10X  -  1  ][110X  then  [10XComputedSylowSubgroups(  [3XG[103X[10X )[ [3Xeven[103X[10X ][110X holds the value of
  [10XSylowSubgroup(  [3XG[103X[10X,  [3Xp[103X[10X  )[110X.  The pairs are sorted in increasing order of [3Xp[103X, in
  particular  at  most  one  Sylow [3Xp[103X subgroup of [3XG[103X is stored for each prime [3Xp[103X.
  This  attribute value is maintained by the function [2XSylowSubgroup[102X ([14X39.13-1[114X),
  which  calls  the  operation [10XSylowSubgroupOp( [3XG[103X[10X, [3Xp[103X[10X )[110X to do the real work, if
  the  prime  [3Xp[103X  cannot be found in the list. So methods that do the real work
  should be installed for [10XSylowSubgroupOp[110X and not for [2XSylowSubgroup[102X ([14X39.13-1[114X).[133X
  
  [33X[0;0YThe  same  mechanism  works  for  other  functions  as well, e.g., for [2XPCore[102X
  ([14X39.11-3[114X), but also for [2XHallSubgroup[102X ([14X39.13-3[114X), where the second argument is
  not a prime but a set of primes.[133X
  
  [33X[0;0Y[2XKeyDependentOperation[102X  declares  the  two  operations  and  the attribute as
  described  above,  with  names  [3Xname[103X,  [3Xname[103X[10XOp[110X, and [10XComputed[110X[3Xname[103X[10Xs[110X, as well as
  tester  and  setter  operations  [10XHas[110X[3Xname[103X  and  [10XSet[110X[3Xname[103X,  respectively. Note,
  however,  that  the  tester is not a filter. [3Xdom-req[103X and [3Xkey-req[103X specify the
  required  filters for the first and second argument of the operation [3Xname[103X[10XOp[110X,
  which  are  needed  to create this operation with [2XDeclareOperation[102X ([14X78.1-5[114X).
  [3Xdom-req[103X  is  also  the  required  filter  for  the  corresponding  attribute
  [10XComputed[110X[3Xname[103X[10Xs[110X.  The  fourth  argument  [3Xkey-test[103X  is in general a function to
  which  the  second  argument  [3Xinfo[103X  of  [10X[3Xname[103X[10X( [3XD[103X[10X, [3Xinfo[103X[10X )[110X will be passed. This
  function can perform tests on [3Xinfo[103X, and raise an error if appropriate.[133X
  
  [33X[0;0YFor   example,   to  set  up  the  three  objects  [2XSylowSubgroup[102X  ([14X39.13-1[114X),
  [10XSylowSubgroupOp[110X,   [10XComputedSylowSubgroups[110X  together,  the  declaration  file
  [11Xlib/grp.gd[111X contains the following line of code.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XKeyDependentOperation( "SylowSubgroup", IsGroup, IsPosInt, "prime" );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  this example, [3Xkey-test[103X has the value [10X"prime"[110X, which is silently replaced
  by a function that tests whether its argument is a prime.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4 := Group((1,2,3,4),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27XSylowSubgroup( s4, 7 );;  ComputedSylowSubgroups( s4 );[127X[104X
    [4X[28X[ 7, Group(()) ][128X[104X
    [4X[25Xgap>[125X [27XSylowSubgroup( s4, 2 );;  ComputedSylowSubgroups( s4 );[127X[104X
    [4X[28X[ 2, Group([ (3,4), (1,4)(2,3), (1,3)(2,4) ]), 7, Group(()) ][128X[104X
    [4X[25Xgap>[125X [27XHasSylowSubgroup( s4, 5 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSetSylowSubgroup( s4, 5, Group(()));; ComputedSylowSubgroups( s4 );[127X[104X
    [4X[28X[ 2, Group([ (3,4), (1,4)(2,3), (1,3)(2,4) ]), 5, Group(()), 7, Group(()) ][128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSylowSubgroup( s4, 6 );[127X[104X
    [4X[28XError, SylowSubgroup: <p> must be a prime[128X[104X
    [4X[28X*[1] <<compiled GAP function>>[128X[104X
    [4X[28X   @ GAPROOT/lib/oper1.g:964[128X[104X
    [4X[28X [2] <<compiled GAP function "SylowSubgroup default method">>[128X[104X
    [4X[28X   @ GAPROOT/lib/oper1.g:1007[128X[104X
    [4X[28X<function "SylowSubgroup default method">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at *stdin*:3[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThus  the  prime  test need not be repeated in the methods for the operation
  [10XSylowSubgroupOp[110X  (which  are  installed  to  do the real work). Note that no
  methods    need    be    installed    for    [2XSylowSubgroup[102X   ([14X39.13-1[114X)   and
  [10XComputedSylowSubgroups[110X. If a method is installed with [2XInstallMethod[102X ([14X78.3-1[114X)
  for  a  wrapper  operation such as [2XSylowSubgroup[102X ([14X39.13-1[114X) then a warning is
  signalled  provided  the  [2XInfoWarning[102X  ([14X7.4-8[114X)  level  is  at  least [10X1[110X. (Use
  [2XInstallMethod[102X ([14X78.3-1[114X) in order to suppress the warning.)[133X
  
  
  [1X85.2 [33X[0;0YIn Parent Attributes[133X[101X
  
  [1X85.2-1 InParentFOA[101X
  
  [33X[1;0Y[29X[2XInParentFOA[102X( [3Xname[103X, [3Xsuper[103X, [3Xsub[103X, [3XAorP[103X ) [32X function[133X
  
  [33X[0;0YThis  section  describes  how you can add new [21Xin parent attributes[121X (see [14X31.8[114X
  and  [14X31.7[114X).  As  an  example, we describe how [2XIndex[102X ([14X39.3-2[114X) and its related
  functions are implemented.[133X
  
  [33X[0;0YThere  are  two  operations  [2XIndex[102X  ([14X39.3-2[114X)  and  [10XIndexOp[110X, and an attribute
  [10XIndexInParent[110X. They are created together as shown below, and after they have
  been  created,  methods  need be installed only for [10XIndexOp[110X. In the creation
  process,  [10XIndexInParent[110X  already  gets  one  default  method  installed  (in
  addition  to  the usual system getter of each attribute, see [14X13.5[114X), namely [10XD
  -> IndexOp( Parent( D ), D )[110X.[133X
  
  [33X[0;0YThe operation [2XIndex[102X ([14X39.3-2[114X) proceeds as follows.[133X
  
  [30X    [33X[0;6YIf  it  is  called  with  the  two  arguments  [3Xsuper[103X  and  [3Xsub[103X, and if
        [10XHasParent(  [3Xsub[103X[10X )[110X and [10XIsIdenticalObj( [3Xsuper[103X[10X, Parent( [3Xsub[103X[10X ) )[110X are [9Xtrue[109X,
        [10XIndexInParent[110X is called with argument [3Xsub[103X, and the result is returned.[133X
  
  [30X    [33X[0;6YOtherwise,  [10XIndexOp[110X  is  called  with  the  same  arguments that [2XIndex[102X
        ([14X39.3-2[114X) was called with, and the result is returned.[133X
  
  [33X[0;0Y(Note  that  it  is in principle possible to install even [2XIndex[102X ([14X39.3-2[114X) and
  [10XIndexOp[110X  methods  for  a  number  of  arguments  different  from  two,  with
  [2XInstallOtherMethod[102X ([14X78.3-2[114X), see [14X13.5[114X).[133X
  
  [33X[0;0YThe  call  of  [2XInParentFOA[102X  declares  the  operations  and  the attribute as
  described  above,  with  names [3Xname[103X, [3Xname[103X[10XOp[110X, and [3Xname[103X[10XInParent[110X. [3Xsuper-req[103X and
  [3Xsub-req[103X  specify  the  required filters for the first and second argument of
  the  operation  [3Xname[103X[10XOp[110X,  which  are  needed  to  create  this operation with
  [2XDeclareOperation[102X  ([14X78.1-5[114X).  [3Xsub-req[103X  is  also  the  required filter for the
  corresponding  attribute  [3Xname[103X[10XInParent[110X;  note that [2XHasParent[102X ([14X31.7-1[114X) is [13Xnot[113X
  required  for  the argument [3XU[103X of [3Xname[103X[10XInParent[110X, because even without a parent
  stored,  [10XParent(  [3XU[103X[10X  )[110X  is  legal,  meaning  [3XU[103X itself (see [14X31.7[114X). The fourth
  argument must be [2XDeclareProperty[102X ([14X13.7-5[114X) if [3Xname[103X[10XInParent[110X takes only boolean
  values  (for  example  in  the  case [10XIsNormalInParent[110X), and [2XDeclareAttribute[102X
  ([14X13.5-4[114X) otherwise.[133X
  
  [33X[0;0YFor  example,  to  set  up  the  three  objects [2XIndex[102X ([14X39.3-2[114X), [10XIndexOp[110X, and
  [10XIndexInParent[110X  together,  the  declaration  file  [11Xlib/domain.gd[111X contains the
  following line of code.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInParentFOA( "Index", IsGroup, IsGroup, DeclareAttribute );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that no methods need be installed for [2XIndex[102X ([14X39.3-2[114X) and [10XIndexInParent[110X.[133X
  
  
  [1X85.3 [33X[0;0YOperation Functions[133X[101X
  
  [33X[0;0YChapter [14X41[114X  and,  in  particular,  the  Section [14X41.1[114X  explain  that  certain
  operations such as [14X41.4[114X), besides their usual usage with arguments [3XG[103X, [3XD[103X, and
  [3Xopr[103X,  can also be applied to an external set ([22XG[122X-set), in which case they can
  be  interpreted  as  attributes.  Moreover,  they can also be interpreted as
  attributes  for permutation groups, meaning the natural action on the set of
  its moved points.[133X
  
  [33X[0;0YThe  definition  of  [14X41.4[114X  says  that  a  method  should  be a function with
  arguments  [3XG[103X,  [3XD[103X,  [3Xgens[103X,  [3Xoprs[103X,  and  [3Xopr[103X,  as  in the case of the operation
  [2XExternalSet[102X  ([14X41.12-2[114X)  when  specified  via  [3Xgens[103X and [3Xoprs[103X (see [14X41.12[114X). All
  other syntax variants allowed for [14X41.4[114X (e.g., leaving out [3Xgens[103X and [3Xoprs[103X) are
  handled by default methods.[133X
  
  [33X[0;0YThe default methods for [14X41.4[114X support the following behaviour.[133X
  
  [31X1[131X   [33X[0;6YIf  the only argument is an external set [3Xxset[103X and the attribute tester
        [10XHasOrbits(  [3Xxset[103X[10X )[110X returns [9Xtrue[109X, the stored value of that attribute is
        returned.[133X
  
  [31X2[131X   [33X[0;6YIf  the  only argument is an external set [3Xxset[103X and the attribute value
        is  not  known,  the  default  arguments are obtained from the data of
        [3Xxset[103X.[133X
  
  [31X3[131X   [33X[0;6YIf  [3Xgens[103X  and  [3Xoprs[103X  are  not  specified,  [3Xgens[103X is set to [10XPcgs( [3XG[103X[10X )[110X if
        [10XCanEasilyComputePcgs(  [3XG[103X[10X  )[110X  is  [9Xtrue[109X,  and  to [10XGeneratorsOfGroup( [3XG[103X[10X )[110X
        otherwise; [3Xoprs[103X is set to [3Xgens[103X.[133X
  
  [31X4[131X   [33X[0;6YThe default value of [3Xopr[103X is [2XOnPoints[102X ([14X41.2-1[114X).[133X
  
  [31X5[131X   [33X[0;6YIn the case of an operation of a permutation group [3XG[103X on [10XMovedPoints( [3XG[103X[10X
        )[110X  via  [2XOnPoints[102X  ([14X41.2-1[114X),  if  the  attribute  tester [10XHasOrbits( [3XG[103X[10X )[110X
        returns [9Xtrue[109X, the stored attribute value is returned.[133X
  
  [31X6[131X   [33X[0;6YThe operation is called as [10X[3Xresult[103X[10X:= Orbits( [3XG[103X[10X, [3XD[103X[10X, [3Xgens[103X[10X, [3Xoprs[103X[10X, [3Xopr[103X[10X )[110X.[133X
  
  [31X7[131X   [33X[0;6YIn  the  case  of an external set [3Xxset[103X or a permutation group [3XG[103X in its
        natural action, the attribute setter is called to store [3Xresult[103X.[133X
  
  [31X8[131X   [33X[0;6Y[3Xresult[103X is returned.[133X
  
  [33X[0;0YThe  declaration  of  operations  that  match  the  above pattern is done as
  follows.[133X
  
  [1X85.3-1 OrbitsishOperation[101X
  
  [33X[1;0Y[29X[2XOrbitsishOperation[102X( [3Xname[103X, [3Xreqs[103X, [3Xusetype[103X, [3XAorP[103X ) [32X function[133X
  
  [33X[0;0Ydeclares an attribute [10Xop[110X, with name [3Xname[103X. The second argument [3Xreqs[103X specifies
  the  list  of required filters for the usual (five-argument) methods that do
  the real work.[133X
  
  [33X[0;0YIf the third argument [3Xusetype[103X is [9Xtrue[109X, the function call [10Xop( xset )[110X will –if
  the  value  of  [10Xop[110X  for [10Xxset[110X is not yet known– delegate to the five-argument
  call of [10Xop[110X with second argument [10Xxset[110X rather than with [10XD[110X. This allows certain
  methods  for  [10Xop[110X  to make use of the type of [10Xxset[110X, in which the types of the
  external  subsets  of  [10Xxset[110X  and  of the external orbits in [10Xxset[110X are stored.
  (This  is used to avoid repeated calls of [2XNewType[102X ([14X13.9-3[114X) in functions like
  [10XExternalOrbits(  xset  )[110X,  which call [10XExternalOrbit( xset, pnt )[110X for several
  values of [10Xpnt[110X.)[133X
  
  [33X[0;0YFor  property  testing  functions such as [2XIsTransitive[102X ([14X41.10-1[114X), the fourth
  argument   [3XAorP[103X   must   be  [2XNewProperty[102X  ([14X13.7-4[114X),  otherwise  it  must  be
  [2XNewAttribute[102X  ([14X13.5-3[114X);  in  the former case, a property is returned, in the
  latter case an attribute that is not a property.[133X
  
  [33X[0;0YFor  example,  to set up the operation [2XOrbits[102X ([14X41.4-2[114X), the declaration file
  [11Xlib/oprt.gd[111X contains the following line of code:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XOrbitsishOperation( "Orbits", OrbitsishReq, false, NewAttribute );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe global variable [10XOrbitsishReq[110X contains the standard requirements[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XOrbitsishReq := [ IsGroup, IsList,[128X[104X
    [4X[28X                  IsList,[128X[104X
    [4X[28X                  IsList,[128X[104X
    [4X[28X                  IsFunction ];[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ywhich are usually entered in calls to [2XOrbitsishOperation[102X.[133X
  
  [33X[0;0YThe  new  operation,  e.g., [2XOrbits[102X ([14X41.4-2[114X), can be called either as [10XOrbits(
  [3Xxset[103X[10X  )[110X  for an external set [3Xxset[103X, or as [10XOrbits( [3XG[103X[10X )[110X for a permutation group
  [3XG[103X, meaning the orbits on the moved points of [3XG[103X via [2XOnPoints[102X ([14X41.2-1[114X), or as[133X
  
  [33X[0;0Y[10XOrbits( [3XG[103X[10X, [3XOmega[103X[10X[, [3Xgens[103X[10X, [3Xacts[103X[10X][, [3Xact[103X[10X] )[110X,[133X
  
  [33X[0;0Ywith  a  group  [3XG[103X,  a  domain  or  list  [3XOmega[103X,  generators  [3Xgens[103X  of [3XG[103X, and
  corresponding  elements  [3Xacts[103X  that  act  on [3XOmega[103X via the function [3Xact[103X; the
  default  of [3Xgens[103X and [3Xacts[103X is a list of group generators of [3XG[103X, the default of
  [3Xact[103X is [2XOnPoints[102X ([14X41.2-1[114X).[133X
  
  [33X[0;0YOnly  methods  for  the five-argument version need to be installed for doing
  the  real work. (And of course methods for one argument in case one wants to
  define a new meaning of the attribute.)[133X
  
  [1X85.3-2 OrbitishFO[101X
  
  [33X[1;0Y[29X[2XOrbitishFO[102X( [3Xname[103X, [3Xreqs[103X, [3Xfamrel[103X, [3Xusetype[103X, [3Xrealenum[103X ) [32X function[133X
  
  [33X[0;0Yis used to create operations like [2XOrbit[102X ([14X41.4-1[114X). This function is analogous
  to  [2XOrbitsishOperation[102X  ([14X85.3-1[114X),  but  for operations [3Xorbish[103X like [10XOrbit( [3XG[103X[10X,
  [3XOmega[103X[10X,  [3Xpnt[103X[10X  )[110X.  Since  the  return values of these operations depend on the
  additional argument [3Xpnt[103X, there is no associated attribute.[133X
  
  [33X[0;0YThe  call  of [2XOrbitishFO[102X declares a wrapper function and its operation, with
  names [3Xname[103X and [3Xname[103X[10XOp[110X.[133X
  
  [33X[0;0YThe  second  argument  [3Xreqs[103X  specifies  the list of required filters for the
  operation [3Xname[103X[10XOp[110X.[133X
  
  [33X[0;0YThe  third  argument  [3Xfamrel[103X is used to test the family relation between the
  second  and  third  argument  of  [10X[3Xname[103X[10X(  [3XG[103X[10X, [3XD[103X[10X, [3Xpnt[103X[10X )[110X. For example, [3Xfamrel[103X is
  [10XIsCollsElms[110X  in the case of [2XOrbit[102X ([14X41.4-1[114X) because [3Xpnt[103X must be an element of
  [3XD[103X.  Similarly, in the call [10XBlocks( [3XG[103X[10X, [3XD[103X[10X, [3Xseed[103X[10X )[110X, [3Xseed[103X must be a subset of [3XD[103X,
  and the family relation must be [2XIsIdenticalObj[102X ([14X12.5-1[114X).[133X
  
  [33X[0;0YThe  fourth  argument  [3Xusetype[103X  serves  the  same  purpose as in the case of
  [2XOrbitsishOperation[102X  ([14X85.3-1[114X).  [3Xusetype[103X  can  also  be  an attribute, such as
  [10XBlocksAttr[110X  or  [10XMaximalBlocksAttr[110X.  In  this  case,  if  only one of the two
  arguments [3XOmega[103X and [3Xpnt[103X is given, blocks with no seed are computed, they are
  stored  as  attribute  values  according  to the rules of [2XOrbitsishOperation[102X
  ([14X85.3-1[114X).[133X
  
  [33X[0;0YIf  the  5th  argument is set to [9Xtrue[109X, the action for an external set should
  use  the  enumerator,  otherwise it uses the [2XHomeEnumerator[102X ([14X41.12-5[114X) value.
  This will make a difference for external orbits as part of a larger domain.[133X
  
  
  [1X85.3-3 [33X[0;0YExample: Orbit and OrbitOp[133X[101X
  
  [33X[0;0YFor example, to setup the function [2XOrbit[102X ([14X41.4-1[114X) and its operation [10XOrbitOp[110X,
  the declaration file [11Xlib/oprt.gd[111X contains the following line of code:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XOrbitishFO( "Orbit", OrbitishReq, IsCollsElms, false, false );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe variable [10XOrbitishReq[110X contains the standard requirements[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XOrbitishReq  := [ IsGroup, IsList, IsObject,[128X[104X
    [4X[28X                  IsList,[128X[104X
    [4X[28X                  IsList,[128X[104X
    [4X[28X                  IsFunction ];[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ywhich are usually entered in calls to [2XOrbitishFO[102X ([14X85.3-2[114X).[133X
  
  [33X[0;0YThe  relation  test  via [3Xfamrel[103X is used to provide a uniform construction of
  the  wrapper  functions  created  by  [2XOrbitishFO[102X  ([14X85.3-2[114X),  in spite of the
  different  syntax  of  the  specific  functions. For example, [2XOrbit[102X ([14X41.4-1[114X)
  admits the calls [10XOrbit( [3XG[103X[10X, [3XD[103X[10X, [3Xpnt[103X[10X, [3Xopr[103X[10X )[110X and [10XOrbit( [3XG[103X[10X, [3Xpnt[103X[10X, [3Xopr[103X[10X )[110X, i.e., the
  second  argument [3XD[103X may be omitted; [2XBlocks[102X ([14X41.11-1[114X) admits the calls [10XBlocks(
  [3XG[103X[10X,  [3XD[103X[10X,  [3Xseed[103X[10X, [3Xopr[103X[10X )[110X and [10XBlocks( [3XG[103X[10X, [3XD[103X[10X, [3Xopr[103X[10X )[110X, i.e., the third argument may be
  omitted. The translation to the appropriate call of [10XOrbitOp[110X or [10XBlocksOp[110X, for
  either operation with five or six arguments, is handled via [3Xfamrel[103X.[133X
  
  [33X[0;0YAs  a  consequence,  there must not only be methods for [10XOrbitOp[110X with the six
  arguments  corresponding  to  [10XOrbitishReq[110X,  but  also  methods for only five
  arguments  (i.e.,  without  [3XD[103X).  Plenty  of  examples  are  contained in the
  implementation file [11Xlib/oprt.gi[111X.[133X
  
  [33X[0;0YIn  order  to  handle  a  few  special cases (currently [2XBlocks[102X ([14X41.11-1[114X) and
  [2XMaximalBlocks[102X  ([14X41.11-2[114X)), also the following form of [2XOrbitishFO[102X ([14X85.3-2[114X) is
  supported.[133X
  
  [33X[0;0Y[10XOrbitishFO( [3Xname[103X[10X, [3Xreqs[103X[10X, [3Xfamrel[103X[10X, [3Xattr[103X[10X )[110X[133X
  
  [33X[0;0YThe  functions  in  question depend upon an argument [3Xseed[103X, so they cannot be
  regarded  as  attributes. However, they are most often called without giving
  [3Xseed[103X,  meaning  [21Xchoose any minimal resp. maximal block system[121X. In this case,
  the result can be stored as the value of the attribute [3Xattr[103X that was entered
  as fourth argument of [2XOrbitishFO[102X ([14X85.3-2[114X). This attribute is considered by a
  call  [10XBlocks(  [3XG[103X[10X,  [3XD[103X[10X,  [3Xopr[103X[10X  )[110X (i.e., without [3Xseed[103X) in the same way as [2XOrbits[102X
  ([14X41.4-2[114X) considers [10XOrbitsAttr[110X.[133X
  
  [33X[0;0YTo  set  this  up,  the  declaration file [11Xlib/oprt.gd[111X contains the following
  lines:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XDeclareAttribute( "BlocksAttr", IsExternalSet );[128X[104X
    [4X[28XOrbitishFO( "Blocks",[128X[104X
    [4X[28X    [ IsGroup, IsList, IsList,[128X[104X
    [4X[28X      IsList,[128X[104X
    [4X[28X      IsList,[128X[104X
    [4X[28X      IsFunction ], IsIdenticalObj, BlocksAttr, true );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnd this extraordinary FOA triple works as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4 := Group((1,2,3,4),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27XBlocks( s4, MovedPoints(s4), [1,2] );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XTester( BlocksAttr )( s4 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XBlocks( s4, MovedPoints(s4) );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XTester( BlocksAttr )( s4 );  BlocksAttr( s4 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[ [ 1, 2, 3, 4 ] ][128X[104X
  [4X[32X[104X
  
