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

S.W. Margolis, J.E. Pin
1992 Journal of Pure and Applied Algebra  
Pin, 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  ... 
doi:10.1016/0022-4049(92)90148-9 fatcat:ktgsgbwilrfg5lngsyefpf4qcy

A topological approach to a conjecture of Rhodes

J.E. Pin
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] .  ... 
doi:10.1017/s0004972700027805 fatcat:gto6egxpjjgvjizcwmha4j53ki

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 Rhodestype II conjecture.  ... 

Reduction theorem for the Type-II conjecture for finite monoids

Karsten Henckell, John Rhodes
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.  ... 
doi:10.1016/0022-4049(90)90048-m fatcat:lnokemiahnb2tjztvyhonhh63y

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 theRhodes 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

Karsten Henckell, John Rhodes
1992 Journal of Algebra  
We prove that if all regular elements 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] ).  ... 
doi:10.1016/0021-8693(92)90141-8 fatcat:xr22vzrpnnhqfonfcljt22dnwe

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]

Stuart W. Margolis
2014 arXiv   pre-print
This paper is 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] .  ... 
arXiv:1409.2308v2 fatcat:f6awhe4vjbg5jmxobgrtj3sifi

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

KARSTEN HENCKELL, STUART W. MARGOLIS, JEAN-ERIC PIN, JOHN RHODES
1991 International journal of algebra and computation  
This paper is concerned with 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.  ... 
doi:10.1142/s0218196791000298 fatcat:bo43mcbkjnfl3mhbqgqnwfl7oy

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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