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<Chapter Label="data"><Heading>The Data in the Library</Heading>
In this chapter we outline how the semigroups in the library were found, exactly what semigroups are available, how they are stored, and how further information regarding the properties of these semigroups is handled.<P/> <!--The main section describes in detail the data structure of the &GAP; library files (<F>*.gl</F> files) in which the information is stored. Three (similar) ways of encoding are used for different types of semigroups.<P/> Besides the semigroups themselves a lot of additional information about their properties is available. Which and how it is stored can be found in the last section of this chapter.--> <!-- ############### --> <Section><Heading>Creation of the Semigroups</Heading> This section describes which semigroups are contained in the library and how they were determined. <P/> The purpose of the library is to provide one semigroup of every `structural type'. The semigroups are represented by their multiplication table. Usually, say, for groups, `structural type' means 'up to isomorphism' which corresponds to relabelling the elements. Roughly speaking, transposing the multiplicationable of a semigroup does not alter its important structure features either. More precisely, the usual description of the structure of a semigroup using Green's relations is invariant under these operations. So, we consider two semigroups to be of the same structure if they are isomorphic or anti-isomorphic. We will refer to semigroups that are isomorphic or anti-isomorphic as <E>equivalent</E>. The vast number of non-equivalent semigroups with small numbers of elements (see Table <Ref Label="nrobjects"/>) limits us to providing the semigroups with at most 8 elements. <P/> The problem of constructing semigroups up to isomorphism and anti-isomorphism has been considered by many authors. For very small orders, that is 1 to 5, all the semigroups up to isomorphism and anti-isomorphism were computed by hand <Cite Key="tamura1"/> and <Cite Key="tamura2"/>. The first instance of the use of computers to find all semigroups up to isomorphism and anti-isomorphism is described in Forsythe <Cite Key="For55"/>. Subsequently, the number of semigroups with 6 elements was found by Plemmons <Cite Key="Ple67"/>, with 7 elements by Jürgensen and Wick <Cite Key="JW77"/>, with 8 by Satoh, Yama and Tokizawa <Cite Key="SZT94"/>, and with 9 by Distler, Kelsey, and Mitchell in 2008. Even if the authors could store their results they had no means to make them publicly available. Plemmons, for example, explicitly states that he had all multiplication tables for semigroups of size 6 on magnetic tape. Jürgensen and Wick back in 1977 did not store the semigroups of size 7 because of their large number. The tables for semigroups with 8 elements use up several gigabytes of disk space (while the compressed library files in <Package>Smallsemi</Package> need only 22 MB). <P/> Trying to recreate the results from the existing literature, it quickly becomes obvious that even some 15 years later, with considerably more computing power available, the task of obtaining all semigroups with 8 elements is still by no means trivial. Our technique was to find all associative multiplication tables up to isomorphism and anti-isomorphism using a combination of &GAP; and the Constraint Satisfaction Problem (CSP) solver Minion <Cite Key="minion"/>. <!--Specific mathematical details can found in <Cite Key=""/> and --> <!--specific details of the actual computation can be found in <Cite --> <!--Key=""/>.--> More specific details on the search will be available in a later version of <Package>Smallsemi</Package>. <P/> </Section> <!-- ############### --> <Section><Heading>Storing the Semigroups</Heading> As discussed in the previous section, we store data relating to the multiplication table of one representative of every class of equivalent semigroups with 1 to 8 elements. <!--We will refer to these representatives as canonical tables. In this section we explain which tables are stored and how. We will refer to multiplication tables that define equivalent semigroups as <E>equivalent</E>.--> <P/> The tables for semigroups with 2 to 7 elements are stored in the files <F>data2.gl.gz</F> to <F>data7.gl.gz</F> in the directory <F>data/data2to7</F>. <P/> For semigroups of size 8 the data is contained in the directories <F>data/data8-3nil</F> and <F>data/data8</F>. The former contains the data relating to 3-nilpotent semigroups (see <Ref Prop="NilpotencyDegree" />) and the latter the data for all the remaining semigroups of size 8. <P/> The tables of non-3-nilpotent semigroups are partitioned into files <F>8diag<entries in the diagonal>.gl.gz</F> with respect to their diagonals. For example, <F>8diag12345678.gl.gz</F> contains tables for all the bands of order 8. <P/> Any 3-nilpotent semigroup has a unique minimal generating set containing those elements that do not appear in the table. We only require the subtable with entries corresponding to the product of two generators, as all other products are zero. Thus if <M>m</M> is the number of generators, we retain information regarding the entries of an <M>m \times m</M> table. However, we do not store all the tables in this case. The <M>m \times m</M> tables can be sorted into ranges and then the first table and the number of tables in the range are stored. For every diagonal there is a file <F>diag<entries in the diagonal>.gl.gz</F> containing the first tables of each range and a separate file named <F>diag<entries in the diagonal>pos.gl.gz</F> containing the lengths of these ranges. <P/> <Table Align="l|l|r|r|r"> <Row> <Item>class</Item><Item>file names</Item><Item>data size</Item><Item>- gzipped</Item><Item>compression factor</Item></Row> <HorLine/> <Row> <Item>sizes 2-7</Item> <Item><F>data<size>.gl</F></Item> <Item>39 MB</Item> <Item>680 KB</Item> <Item>58</Item> </Row> <Row> <Item>size 8, not 3-nilpotent</Item> <Item><F>8diag<diagonal>.gl</F></Item> <Item>613 MB</Item> <Item>10 MB</Item> <Item>61</Item> </Row> <Row> <Item>size 8, 3-nilpotent</Item> <Item><F>diag<diagonal>.gl</F></Item> <Item>974 MB</Item> <Item>11 MB</Item> <Item>89</Item> </Row> </Table> All together the &GAP; library files take just under 22 MB of disk space after compression while allowing fast recovering of the data. The compression rates demonstrated in the table above were achieved using <C>gzip</C> with the highest possible compression (-9 switch) as well as careful analysis and intensive testing of how best to structure the data in the files.<P/> The semigroups in the library satisfying certain standard properties have been identified and this information is stored in the files <F>info1.g.gz</F> to <F>info8.g.gz</F>. To find out what properties have been considered see <Ref Var="PrecomputedSmallSemisInfo"/>. </Section> </Chapter> ¤ Dauer der Verarbeitung: 0.17 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28