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New results on the conjecture of Rhodes and on the topological conjecture

1992
*
Journal of Pure and Applied Algebra
*

Pin,

doi:10.1016/0022-4049(92)90148-9
fatcat:ktgsgbwilrfg5lngsyefpf4qcy
*New*results on*the**conjecture**of**Rhodes*and on*the*topological*conjecture*, Journal*of*Pure and Applied Algebra 80 (1992) 305-313. 0022-4049/92/$05.00 0 1992 -Elsevier Science Publishers B.V. ...*The**Conjecture**of**Rhodes*, originally called*the*'*type**II**conjecture*' by*Rhodes*, gives an algorithm to compute*the*kernel*of**a*finite semigroup. ... Thus (3) implies (1) and (1)*A*reduction result In this section, we give*a*simple*proof**of**a*theorem*of*Henckell and*Rhodes*[5] which shows that*the**conjecture**of**Rhodes*can be reduced to*the*...##
###
A topological approach to a conjecture of Rhodes

1988
*
Bulletin of the Australian Mathematical Society
*

*The*"

*type*

*II*

*conjecture*", proposed by J.

*Rhodes*, gives an algorithm to compute

*the*kernel

*of*

*a*given finite semigroup. ... Recently,

*Rhodes*has offered $100 for

*the*solution

*of*his "

*type*

*II*

*conjecture*" [15] , which gives an algorithm to compute

*the*"kernel"

*of*

*a*semigroup. ... We can now state

*the*"

*type*

*II*

*conjecture*", for

*the*solution

*of*which

*Rhodes*has offered $100 [14] . ...

##
###
Page 7031 of Mathematical Reviews Vol. , Issue 94m
[page]

1994
*
Mathematical Reviews
*

Brunetto Piochi (1-SIN; Siena)
94m:20120 20M10
Henckell, Karsten (1-SFLNC-NS; Sarasota, FL)
Blockgroups = powergroups:

*a*consequence*of*Ash’s*proof**of**the**Rhodes**type**II**conjecture*. ... Algebra Com- 20M Semigroups 94m:20124 put. 1 (1991), no. 1, 127-146; MR 92k:20114] has solved*Rhodes*’*type**II**conjecture*. ...##
###
Reduction theorem for the Type-II conjecture for finite monoids

1990
*
Journal of Pure and Applied Algebra
*

*The*

*Type*-

*II*

*conjecture*(that Mrt,=Mrr) is still open. ... We also give

*a*simple

*proof*

*of*Pin's theorem that his topological

*conjecture*implies Mu=&. ... Birget and Peter Jones for reading

*a*draft

*of*this paper and offering suggestions and corrections. ...

##
###
Page 6 of Mathematical Reviews Vol. , Issue 94d
[page]

1994
*
Mathematical Reviews
*

Denote by D(M)

*the*smallest submonoid M containing E(M) and closed under weak conjugation.*The**Rhodes**type**II**conjecture*(strong ... One*of*these has come to be called*the*“*Rhodes**type**II**conjecture*”. We assume here that all monoids are finite with*the*exceptions*of*free monoids and free groups. ...##
###
Page 5700 of Mathematical Reviews Vol. , Issue 90J
[page]

1990
*
Mathematical Reviews
*

*The*set

*of*all

*type*

*II*elements

*of*M is called

*the*kernel

*of*M and is denoted by K(M).

*Rhodes*

*conjecture*(weak form): There exists an algorithm to compute

*the*kernel

*of*

*a*given finite monoid. ... In

*a*series

*of*papers

*the*author

*of*

*the*survey has developed

*a*topological approach to

*the*

*Rhodes*

*conjecture*. These results are also outlined in

*the*article. ...

##
###
Type-II conjecture is true for finite I-trivial monoids

1992
*
Journal of Algebra
*

We prove that if all regular elements

doi:10.1016/0021-8693(92)90141-8
fatcat:xr22vzrpnnhqfonfcljt22dnwe
*of**the*finite monoid M are*type*-*II*eIements (*a*decidable condition which includes d-triviat), then*the*strong*type*-11*conjecture*is true for M, I.e., M,=M,F, and so ... As an application we prove that for P(G),*the*monoid*of*nonempty subsets*of**a*finite group G, [P(G)]"= [P(G)],r holds. ... In 1972*Rhodes*and Tilson [5] proved that*the*strong*type*-*II**conjecture*is true for finite regular monoids (for*a*better*proof*see [S] ). ...##
###
Page 4215 of Mathematical Reviews Vol. , Issue 92h
[page]

1992
*
Mathematical Reviews
*

(pp. 441-452); Karsten Henckell and John

*Rhodes*,*The*theorem*of*Knast,*the*PG = BG and*type*-*II**conjectures*(pp. 453-463); Howard Straubing, Automata, logic and computational complexity (pp. 467-492); ... Jr.,*A*structure theory for finite regu- lar semigroups (pp. 403-423); David Easdown, Birordered sets:*a*tool for constructing semigroups (pp. 424-437); Karsten Henckell,*The**type*-*II**conjecture*:*a*survey ...##
###
The q-theory of finite semigroups: history and mathematics
[article]

2014
*
arXiv
*
pre-print

This paper is

arXiv:1409.2308v2
fatcat:f6awhe4vjbg5jmxobgrtj3sifi
*a*historical and mathematical review*of**the*book, "*The*q-theory*of*Finite Semigroups" by John*Rhodes*and Benjamin Steinberg. ...*The**type**II**conjecture*stated that*the**type**II*subsemigroup*of**a*finite semigroup S is*the*smallest subsemigroup T*of*S containing*the*idempotents and such that if sts = s, s, t ∈ S, then sT t ∪ tT s ⊆ ...*The*statement and*proof*involve preliminary material on inevitable substitutions, an idea that goes back to Ash's original solution*of**the**type**II**conjecture*[6] . ...##
###
ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

1991
*
International journal of algebra and computation
*

This paper is concerned with

doi:10.1142/s0218196791000298
fatcat:bo43mcbkjnfl3mhbqgqnwfl7oy
*the*many deep and far reaching consequences*of*Ash's positive solution*of**the**type**II**conjecture*for finite monoids. ... In particular, we show that*the**type**II**conjecture*is equivalent with two other*conjectures*on*the*structure*of*closed sets (one*conjecture*for*the*free monoid and another one for*the*free group). ... This weak form*of**the**type**II**conjecture*was also solved by Ash as*a*warm up to his future*proof**of**the*full*conjecture*. ...##
###
Page 7999 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

One

*of**the*most important results*of*finite semigroup theory is*the**Rhodes**type*I*conjecture*, involving an algorithm io compute*the*kernel*of**a*finite monoid. ... Algebra Comput. 9 (1999), no. 3-4, 241-261; MR 2001a:20102] introduced*a*concept called hyperdecidability*of**a*pseudovariety*of*semigroups, and showed that Ash’s*proof**of**the**type**II**conjecture*proves ...##
###
Page 4563 of Mathematical Reviews Vol. , Issue 84k
[page]

1984
*
Mathematical Reviews
*

“In part

*II**the*errors*of*three*types**of*interpolation splines are given. Let f(x) be*a*function*of*C'[*a*,b], /=3,4,5, and let s(x) be*the*quartic spline*of*interpolation to f(x). ... It is*the*purpose*of*this talk to survey*the*current status*of*this*conjecture*, to correct*a*mistake in*the*verification*of**the*case k=3 in one*of*our earlier papers [J. Approx. ...##
###
Page 1696 of Mathematical Reviews Vol. , Issue 2002C
[page]

2002
*
Mathematical Reviews
*

Verification

*of*this*conjecture*would imply*the*validity*of**the*famous*Rhodes**type**II**conjecture*in full generality. ...*The**proof*is independent*of**the*Henkel-*Rhodes*theorem, and even when restricted to this special case gives*the*shortest least technical*proof*at this stage. ...##
###
Page 19 of Mathematical Reviews Vol. , Issue 94c
[page]

1994
*
Mathematical Reviews
*

Hall,

*A*concept*of*variety for regular semigroups (101-115); Karsten Henckell, Blockgroups = powergroups:*a*consequence*of*Ash’s*proof**of**the**Rhodes**type**II**conjecture*(117-134); John M. ... ., Some properties*of*S(R) (163-179); Stuart W. Margolis, Consequences*of*Ash’s*proof**of**the**Rhodes**type**II*con- jecture (180-205); John Meakin and Mark Sapir [M. V. ...##
###
Page 938 of Mathematical Reviews Vol. , Issue 2003B
[page]

2003
*
Mathematical Reviews
*

*The*inverse monoid-theoretic characterizations are motivated by

*the*work which surrounds

*the*

*Rhodes*

*type*

*II*

*conjecture*[see K. Henck- ell et al., Internat. J. ... Among

*the*most interesting applications (for

*the*reviewer) are Premet’s

*proof*

*of*

*the*Kac-Weisfeiler

*conjecture*[

*A*.

*A*. Premet, Invent. ...

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