On Semigroups Amalgams
Kuupäev
2013
Autorid
Ajakirja pealkiri
Ajakirja ISSN
Köite pealkiri
Kirjastaja
Tartu Ülikool
Abstrakt
Consider two groups G1 and G2 intersecting in a common subgroup U. Can
we find a group W such that G1 and G2 are contained in it and their intersection
is still U? More formally, one asks if the group amalgam [U;G1,G2]
is embeddable. This question (about the embeddability of group amalgams)
was first posed by Otto Schreier. In 1927 he proved that all group amalgams
are embeddable. It was then natural to ask if this result could be expanded
to the class of semigorups. In 1957 N. Kimura constructed a counter example
in his doctoral thesis, showing that this is not the case for semigroups. J. M.
Howie proved in 1962 that a semigroup amalgam [U;S1, S2], in which S1 and
S2 are groups, is embeddable if and only if U is also a group. Howie’s result
was particularly interesting as it provided an infinite class of non-embeddable
semigroup amalgams. In 1975 T. E. Hall generalized Schreier’s result to the
class of inverse semigroups (Journal of Algebra 34, 375–385 (1975)). The aim
of this thesis is to consider some situations where semigroup amalgams fail
to embed. In the first chapter frequently used terminology and some needed
lemmas are introduced. In the second chapter we discuss the existence of
pushouts, a categorical notion that is linked with amalgamation. The last
chapter, which concentrates on semigroup amalgams, provides some results
regarding non-embeddable semigroup amalgams and establishes a link between
pushouts and amalgamation.
Kirjeldus
Antud lõputöö käsitleb poolrühma amalgaame. Poolrühma amalgaamide
all mõistame listi [U;S1, S2; !1,!2], kus U, S1 ja S2 on poolr¨uhmad ning
!i ∶ U → Si (i ∈ {1, 2}) on monomorfismid. ¨ Oeldakse, et poolr¨uhma amalgaam
[U;S1, S2; !1,!2] on sisestatav poolr¨uhma W, kui esiteks leiduvad monomorfismid
μi ∶ S1 → W (i ∈ {1, 2}) nii, et diagramm
U
!2
✏
!1 /S1
μ1
✏
S2 μ2
/W
kommuteerub. Ja teiseks juhul, kui mingid kaks elementi poolr¨uhmadest S1
ja S2 kujutatakse poolr¨uhmas W samaks elemendiks, siis leidub neil ¨uhine
originaal poolr¨uhmas U.
O. Schreier p¨ustitas esimesena k¨usimuse, kas r¨uhma amalgaamid on sisestatavad
r¨uhma. Aastal 1927 t˜oestas ta, et see t˜oesti nii on. Seega oli
loomulik k¨usida, kas sellist omadust oleks v˜oimalik edasi kanda ka teistele
klassidele. Aastal 1957 konstrueeris N. Kimura oma doktorit¨o¨os kontran¨aite,
mis t˜oestas, et k˜oik poolr¨uhma amalgaamid ei ole sisestatavad poolr¨uhma.
J. M. Howie t˜oestas aastal 1962, et poolr¨uhma amalgaam [U;S1, S2], kus S1
ja S2 on r¨uhmad, on sisestatav siis ja ainult siis kui U on samuti r¨uhm. See
tulemus oli eriti huvitav, kuna t¨anu sellele tekkis l˜opmatu hulk poolr¨uhma
amalgaame, mis ei ole sisestatavad. T. E. Hall t˜oestas aastal 1975, et Schreieri
tulemust on v˜oimalik ¨uldistada ning et analoogne tulemus kehtib ka inverssete
poolr¨uhmade korral.
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Antud l˜oput¨o¨o eesm¨argiks oli l¨ahemalt tutvuda poolr¨uhma amalgaamidega
ning t˜oestada m˜oned tulemused poolr¨uhma amalgaamide sisestusest. Viimases
peat¨ukis on t˜oestatud, et poolr¨uhma amalgaam [U;S1, S2], kus S1 ja
S2 on kommutatiivsed regulaarsed poolr¨uhmad, on sisestatav parajasti siis,
kui U on regulaarne. See tulemus ¨uldistab teatud m¨a¨aral varem Howie poolt
t˜oestatud tulemust, et kommutatiivne regulaarne amalgaam, kus ka U on
regulaarne, on sisestatav. Samuti on ka t˜oestatud, et poolr¨uhma amalgaam,
kus S1 ja S2 on t¨aielikult regulaarsed ei ole sisestatav, kui U ei ole t¨aielikult
regulaarne. Sama tulemus t˜oestatakse ka Cli↵ordi poolr¨uhmade jaoks.
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Märksõnad
semigroups, semigroup amalgams, pushouts, embedding, bachelor thesis