The semigroups of sizes 1 to 8 are available up to isomorphism and
anti-isomorphism in <Package>Smallsemi</Package>. Every semigroup in the
library is identified by its size <M>m</M> and a number <M>n</M> lying between
1 and the number of semigroups of size <M>m</M> (see Table <Ref
Label="nrobjects"/>). We call the pair <M>(m,n)</M> the <E>ID</E> of the
semigroup.
<P/>
In this section we give details about the functions relating to individual
semigroups in <Package>Smallsemi</Package>. This includes how to access
semigroups in the library and how to identify the semigroup in the library
equivalent to an arbitrary semigroup (of size
1 to 8).
<P/>
If you are interested in the properties of a semigroup in the library or would
like to find all the semigroups satisfying a given set of properties please see
Section <Ref Sect="props"/> or Section <Ref Sect="enums"/> respectively.
<P/>
<Section Label="props"><Heading>Properties of Semigroups</Heading>
In this section we detail the &GAP; functions that can be used to determine
whether a small semigroup satisfies a certain property. Let <A>S</A> be a
semigroup. Then
<List>
<Item>
<A>S</A> is a <E>left zero semigroup</E> if <A>xy=x</A> for all
<A>x,y</A> in <A>S</A>.
</Item>
<Item>
<A>S</A> is a <E>right zero semigroup</E> if <A>xy=y</A> for all
<A>x,y</A> in <A>S</A>.
</Item>
<Item>
<A>S</A> is <E>commutative</E> if <A>xy=yx</A> for all <A>x,y</A> in
<A>S</A>.
</Item>
<Item>
<A>S</A> is <E>simple</E> if it has no proper two-sided ideals.
</Item>
<Item>
<A>S</A> is <E>zero simple</E> if the only 2-sided ideals are
<A>{0}</A> and <A>S</A>.
</Item>
<Item>
<A>S</A> is <E>regular</E> if for all <A>x</A> in <A>S</A> there exists
<A>y</A> in <A>S</A> such that <A>xyx=x</A>.
</Item>
<Item>
<A>S</A> is <E>completely regular</E> if every element of <A>S</A>
lies in a subgroup.
</Item>
<Item>
<A>S</A> is an <E>inverse semigroup</E> if every element <A>x</A> in
<A>S</A> has a unique semigroup inverse, that is, a unique element
<A>y</A> such that <A>xyx=x</A> and <A>yxy=y</A>.
</Item>
<Item>
<A>S</A> is a <E>Clifford semigroup</E> if it is a regular semigroup
whose idempotents are central, that is, for all <A>e,x</A> in <A>S</A>
where <A>e^2=e</A> we have that <A>ex=xe</A>.
</Item>
<Item>
<A>S</A> is a <E>band</E> if every element is an idempotent, that is,
<A>x^2=x</A> for all <A>x</A> in <A>S</A>.
</Item>
<Item>
<A>S</A> is a <E>Brandt semigroup</E> if it is inverse and zero simple.
</Item>
<Item>
<A>S</A> is a <E>rectangular band</E> if for all <A>x,y,z</A> in <A>S</A>
we have that <A>x^2=x</A> and <A>xyz=xz</A>.
</Item>
<Item>
<A>S</A> is a <E>semiband</E> if it is generated by its idempotent
elements, that is, elements satisfying <A>x^2=x</A>.
</Item>
<Item>
<A>S</A> is an <E>orthodox semigroup</E> if it is regular and its
idempotents (elements satisfying <A>x^2=x</A>) form a subsemigroup.
</Item>
<Item>
<A>S</A> is a <E>zero semigroup</E> if there exists an element <A>0</A>
in <A>S</A> such that <A>xy=0</A> for all <A>x,y</A> in <A>S</A>.
</Item>
<Item>
<A>S</A> is a <E>zero group</E> if there exists an element <A>0</A> in
<A>S</A> such that <A>S</A> without <A>0</A> is a group and for all
<A>x</A> in <A>S</A> we have that <A>x0=0x=0</A>.
</Item>
</List>
The <Package>MONOID</Package> package was used to determined which
semigroups in the library satisfy the properties above. All of the
resulting information is stored in the library.
<Section Label="coclass"><Heading>Nilpotent semigroups by coclass</Heading>
A useful parameter in the classification of nilpotent semigroups is their
coclass. For a finite nilpotent semigroup of order <M>n</M> and nilpotency
degree <M>d</M> the coclass is defined as <M>n-d</M>. In <Cite Key="Dis14"/>
lists up to (anti-)isomorphism are provided for nilpotent semigroups of coclass
0, 1, and 2. The semigroups in the lists are given by finite presentations. In
this section we describe a function that allows to access such lists in &GAP;.
<P/> A further invariant of a nilpotent semigroup <M>S</M> is the size of its
unique minimal generating set <M>S\backslash S^2</M>. The possible sizes for a
particular coclass are restricted. Monogenic nilpotent semigroups are precisely
those of coclass 0. For coclass <M>d \geq 1</M> the size of the minimal
generating set is at least 2 and at most <M>d+1</M>.
In this section functionality around the starred Green's relations is
described. The five starred Green's relations are R^*,
<M>L^*</M>, <M>J^*</M>, <M>H^*</M>, and <M>D^*</M>;
two elements <M>a</M>, <M>b</M> from a semigroup <M>S</M> are
<M>R^*</M>-related if for all <M>x, y \in S^1: xa=ya</M> if and only if
<M>xb = yb</M>; and <M>a</M> and <M>b</M> are <M>L^*</M>-related if
for all <M>x, y \in S^1:
ax=ay</M> if and only if <M>bx = by</M>. In parallel to the classical
Green's relations
<M>H^*=R^* \wedge L^*</M> and
<M>D^*= R^* \vee L^*</M> (but <M>R^* \circ L^* = L^* \circ R^*</M>
does <E>not</E> hold in general). To describe <M>J^*</M> is a bit
more technical. For <M>a,b \in S</M> one can show that <M>b</M> lies in
<M>J^*(a)</M>, the principal *-ideal of <M>a</M>, if and only if there
exist <M>c_0,c_1\ldots,c_n\in S</M> and <M>x_1,\ldots,x_n,y_1,\ldots,
y_n\in S^1</M> such that <M>a=c_0, b=c_n</M> and
<M>c_iD^*x_ic_{i-1}y_i</M> for <M>1\leq i\leq n</M>. Then <M>aJ^*b</M>
if and only if both <M>a\in J^*(b)</M> and <M>b\in J^*(a)</M>
<P/>
Note that even for finite semigroups <M>J^*</M> does not always equal
<M>D^*</M> (in contrast to the situation for classical Green's
relations). Using <Package>Smallsemi</Package> it was shown that there
exist semigroups of order 8 with <M>J^*\neq D^*</M>
<Cite Key="DMU13"/>.
<Section Label="enums"><Heading>Families of Semigroups</Heading>
In this section we describe how to find semigroups in the library
satisfying a given set of parameters.<P/>
The following functions have the same usage but may return different
values: <Ref Func="EnumeratorOfSmallSemigroups"/>,
<Ref Func="AllSmallSemigroups"/>,
<Ref Func="IdsOfSmallSemigroups"/>,
<Ref Func="IteratorOfSmallSemigroups"/>,
<Ref Func="NrSmallSemigroups"/>,
<Ref Func="OneSmallSemigroup"/>,
<Ref Func="PositionsOfSmallSemigroups"/>,
<Ref Func="RandomSmallSemigroup"/>. <P/>
The number of arguments should be odd:
<List>
<Item>
the first argument <C>arg[1]</C> should be a positive integer, a list
of positive integers, or an enumerator or iterator of small semigroups
satisfying <Ref Func="IsEnumeratorOfSmallSemigroups"/> or
<Ref Func="IsIteratorOfSmallSemigroups"/>
</Item>
<Item>
the even arguments <C>arg[2i]</C>, if present, should be a function
</Item>
<Item>
the odd arguments <C>arg[2i+1]</C> argument should be a possible value
that can be returned by the function <C>arg[2i]</C>.
</Item>
</List>
In the case that the function is <Ref Func="AllSmallSemigroups"/> and
<C>arg[1]</C> is a positive integer, then the returned value is a list of
all semigroups <M>S</M> with <C>arg[1]</C> elements such that
<C>arg[2i](S)=arg[2i+1]</C>.<P/>
For example, to obtain all the commutative semigroups with <A>3</A>
idempotents of sizes <A>2</A> to <A>5</A> use one of <Ref
Func="EnumeratorOfSmallSemigroups"/>, <Ref Func="AllSmallSemigroups"/>,
<Ref Func="IteratorOfSmallSemigroups"/> with argument
<Log>
[2 .. 5], IsCommutative, true, Is3IdempotentGenerated, true
</Log>
<Ref Func="AllSmallSemigroups"/> returns a list of all such semigroups,
<Ref Func="EnumeratorOfSmallSemigroups"/>, and
<Ref Func="IteratorOfSmallSemigroups"/> return an enumerator and an iterator of
all such semigroups, respectively. For more information on enumerators and
iterators see <Ref Attr="Enumerator" BookName="ref"/>, or
<Ref Oper="Iterator" BookName="ref"/>. The following are rules of thumb
regarding the different situations when these functions should be used in
order of slowest to fastest and greatest memory use to least:
<List>
<Item>
<Ref Func="AllSmallSemigroups"/> should be used if the number of
semigroups is not too large and you want to keep the created semigroups
in a list.
</Item>
<Item>
<Ref Func="EnumeratorOfSmallSemigroups"/> should be used when the
functions in even indexed positions are those stored in the library
(see <Ref Var="PrecomputedSmallSemisInfo"/>) or you want to repeatedly
search the same set of semigroups and there are too many to store in a
list. Note that the enumerator stores the id numbers of its elements
but not the semigroups themselves. Hence every time an element of the
enumerator is required it must be recreated from the multiplication
table data.
</Item>
<Item>
<Ref Func="IteratorOfSmallSemigroups"/> should be used if the functions
in even indexed positions are not stored in the library (see <Ref
Var="PrecomputedSmallSemisInfo"/>) or if you just want to run
through all the semigroups satisfying the specified parameters once
only. Note that each new call of <Ref Func="IteratorOfSmallSemigroups"/>
requires &GAP; to recompute its elements which may be slow if the
functions are user-defined or not stored in the library.
</Item>
</List>
Further information on the relative virtues of these different commands can
be found in Chapter <Ref Chap="examples"/>.<P/>
As a further example, if we want to obtain a single non-simple semigroup
with <M>7</M> elements and trivial automorphism group, then we would use
one of the functions <Ref Func="OneSmallSemigroup"/> or <Ref
Func="RandomSmallSemigroup"/> with argument<P/>
<Log>
7, IsSimpleSemigroup, false, x -> IsTrivial(AutomorphismGroup(x)), true
</Log>
<Ref Func="OneSmallSemigroup"/> should return an answer more quickly than
<Ref Func="RandomSmallSemigroup"/>. Also note that <Ref
Func="OneSmallSemigroup"/> will always return the same semigroup, i.e. the
first semigroup in the library with the given parameters.
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