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UID:7641nb53k7v4p2spd9ui7dfg9k@google.com
CATEGORIES:{lang hu}Algebra szeminárium{/lang}{lang en}Algebra seminar{/lang}
SUMMARY:Igor Dolinka (University of Novi Sad): The word problem for one-relator inverse monoids: New developments
LOCATION:Bolyai Intézet, I. emelet, Riesz terem, Aradi vértanúk tere 1., Szeged
DESCRIPTION;ENCODING=QUOTED-PRINTABLE:\nAbstract. In early 1930s, W. Magnus proved his classical result that the
word problem is decidable for all one-relator groups (groups given by one d
efining relation). This result is based on another important theorem proved
earlier by Magnus, the Freiheitssatz, which, roughly speaking, locates man
y free subgroups in one-relator groups. This inspired investigations of the
word problem for other algebraic structures defined by a single relation.
For example, in the 1960s Shirshov proved that the word problem is decidabl
e for all one-relator Lie algebras. Surprisingly, the problem whether the w
ord problem is decidable for all one-relator monoids is still open, althoug
h several important cases have been resolved by Adjan in 1966, and Adyan an
d Oganessyan in 1987.\nAn important intermediate class of algebraic structu
res lying between groups and monoids are that of inverse monoids. In 2001 I
vanov, Margolis and Meakin highlighted the importance of investigating one-
relator inverse monoids by showing that the (conjectured) decidability of t
he word problem for one-relator special inverse monoids (in fact, for a par
ticular class of these inverse monoids) would imply a positive solution of
the word problem for all one-relator monoids. In this talk, I will present
two major recent contributions to this topic:\n(1) A result of R.D. Gray sh
owing that the word problem for one-relator special inverse monoids is unde
cidable in its full generality; furthermore, there exists a one-relator gro
up with undecidable submonoid membership problem.\n(2) The joint results of
the speaker and R. D. Gray pertaining to the so-called prefix membership p
roblem for one-relator groups, immediately implying decidability of the wor
d problem for wide classes of one-relator special inverse monoids.\nAlong t
he way, I will explain the relation of the word problem for one-relator inv
erse monoids to the word problem of one-relator monoids, and also to severa
l problems in group theory such as the prefix membership problem for one-re
lator groups and the role of embeddability of right-angled Artin groups.
DTSTAMP:20210514T073450Z
DTSTART;TZID=Europe/Budapest:20191204T100000
DTEND;TZID=Europe/Budapest:20191204T120000
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