Abstract
This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups might possibly satisfy. The main theorem states that given a finite collection of pseudovarieties, each satisfying certain properties of the sort alluded to above, any iterated semidirect product of these pseudovarieties is decidable. In particular, the pseudovariety G of finite groups satisfies these properties. J. Rhodes has announced a proof, in collaboration with J. McCammond, that the pseudovariety A of finite aperiodic semigroups satisfies these properties as well. Thus, our main theorem would imply the decidability of the complexity of a finite semigroup. Their work, in light of our main theorem, would imply the decidability of the complexity of a finite semigroup.
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References
D. Albert, R. Baldinger, and J. Rhodes, The identity problem for finite semi-groups (the undecidability of), J. Symbolic Logic, 57 (1992), 179–192.
J. Almeida, Implicit operations on finite J-trivial semigroups and a conjecture of I. Simon, J. PureAppl. Algebra, 69 (1990), 205–218.
J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Sin-gapore, 1995 (English translation).
J. Almeida, Hyperdecidable pseudovarieties and the calculation of semidirect products, Internat. J. Algebra Comput., 9 (1999), 241–261.
J. Almeida, On hyperdecidable pseudovarieties of simple semigroups, Inter- nat. J. Algebra Comput., 1998, to appear.
J. Almeida, On a problem of Brzozowski and Fich, in Semigroups and Appli- cations, J. M. Howie and N. Ruskuc, eds., World Scientific, Singapore, 1998, 1–17.
J. Almeida and A. Azevedo, Globals of pseudovarieties of commutative semi-groups: The finite basis problem, decidability, and gaps, Technical Report CMUP 1999–10, University of Porto, Porto, Portugal, 1999.
J. Almeida, A. Azevedo, and L. Teixeira, On finitely based pseudovarieties of the forms V * D and V * Dn J. PureAppl. Algebra,1998, to appear.
J. Almeida, A. Azevedo, and M. Zeitoun, Pseudovariety joins involving,trivial semigroups and completely regular semigroups, Internat. J. Algebra Comput., 9 (1999), 99–112.
J. Almeida and A. Escada, On the equation V * G = εV, Technical Report CMUP 98–6, University of Porto, Porto, Portugal, 1998.
J. Almeida and P. V. Silva, SC-hyperdecidability of R, Theoret. Comput. Sci., 1998, to appear.
J. Almeida and B. Steinberg, Iterated semidirect products with applications to complexity, Proc. London Math. Soc., 1998, to appear.
J. Almeida and P. Weil, Relatively free profinite monoids: An introduction and examples, in Semigroups, Formal Languages and Groups, vol. 466, J. B. Fountain, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995, 73–117.
J. Almeida and P. Weil, Free profinite semigroups over semidirect products, Russian Math. (Iz. VUZ), 39 (1995), 1–27.
J. Almeida and P. Weil, Profinite categories and semidirect products, J. Pure AppL Alge-bra,123 (1998), 1–50.
J. Almeida and M. Zeitoun, The pseudovariety J is hyperdecidable, Theoret. Inform. Appl., 31 (1997), 457–482.
C. J. Ash, Inevitable graphs: A proof of the type II conjecture and some related decision procedures, Internat. J. Algebra Comput., 1 (1991), 127–146.
J. C. Birget and J. Rhodes, Almost finite expansions of arbitrary semigroups, J. PureAppl. Algebra, 32 (1984), 127–146.
S. Burris and H. P. SankappanavarA Course in Universal Algebra, Springer-Verlag, New York, 1981.
M. Delgado, Teorema do tipo II e hiperdecidibilade de pseudovariedades de grupos, Ph.D. thesis, University of Porto, Porto, Portugal, 1997.
S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York, 1976.
T. Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc., 1 (1969), 399–403.
M. D. Fried and M. Jarden, Field Arithmetic, Springer-Verlag, Berlin, 1986.
K. Henckell, Pointlike sets: The finest aperiodic cover of a finite semigroup, J. PureAppl. Algebra, 55 (1988), 85–126.
K. Henckell, Product expansions, J. PureAppl. Algebra, 101 (1995), 157–170.
K. Henckell, S. Margolis, J.-E. Pin, and J. Rhodes, Ash’s type II theorem, profinite topology and Malcev products, part I, Internat. J. Algebra Comput., 1 (1991), 411–436.
K. Krohn and J. Rhodes, Algebraic theory of machines I: Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc., 116 (1965), 450–464.
P. R. Jones, Profinite categories, implicit operations, and pseudovarieties of categories, J. Pure Appl. Algebra, 109 (1995), 61–95.
S. MacLane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.
J. McCammond, The solution to the word problem for the relatively free semigroups satisfying T° = T a+b with a ≥ 6, Internat. J. Algebra Comput., 1 (1991), 1–32.
J.-E. Pin, Varieties of Formal Languages, Plenum, New York, 1986.
J. Reiterman The Birkhoff theorem for finite algebrasAlgebra Universalis, 14 (1982), 1–10.
J. Rhodes, Kernel systems: A global study of homomorphisms on finite semi-groups, J. Algebra, 49 (1977), 1–45.
J. Rhodes, Undecidability, automata and pseudovarieties of finite semigroups, Internat. J. Algebra Comput., 9 (1999), 455–473.
J. Rhodes and B. Steinberg, Pointlikes sets, hyperdecidability, and the identity problem for finite semigroups, Internat. J. Algebra Comput., 9 (1999), 475–481.
L. Ribes and P. A. Zalesskii, On the profinite topology on a free group, Bull. London Math. Soc., 25 (1993), 37–43.
L. Ribes and P. A. Zalesskii, The pro-p topology of a free group and algorithmic problems in semigroups, Internat. J. Algebra Comput., 4 (1994), 359–374.
I. Simon, Hierarchies of events of dot-depth one, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada, 1972.
B. Steinberg, On pointlike sets and joins of pseudovarieties, Internat. J. Algebra Comput., 8 (1998), 203–231.
B. Steinberg, On algorithmic problems for joins of pseudovarieties, Semigroup Forum, 1998, to appear.
B. Steinberg, Semidirect products of categories and applications, J. Pure and Applied Algebra, 142 (1999), 153–182.
B. Steinberg, Inevitable graphs and profinite topologies: Some solutions to al-gorithmic problems in monoid and automata theory stemming from group theory, Internat. J. Algebra Comput., 1998, to appear.
B. Steinberg, A delay theorem for pointlikes, Semigroup Forum, 1999, to appear.
B. Tilson, Categories as algebra: An essential ingredient in the theory of monoids, J. Pure Appl. Algebra, 48 (1987), 83–198.
P. G. Trotter and M. V. Volkov, The finite basis problem in the pseudovariety joins of aperiodic semigroups with groups, Semigroup Forum, 52 (1996), 83–91.
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Almeida, J., Steinberg, B. (2000). Syntactic and Global Semigroup Theory: A Synthesis Approach. In: Birget, JC., Margolis, S., Meakin, J., Sapir, M. (eds) Algorithmic Problems in Groups and Semigroups. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1388-8_1
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DOI: https://doi.org/10.1007/978-1-4612-1388-8_1
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