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Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

  • Samuel J. van Gool (a1) and Benjamin Steinberg (a1)


This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$ , $T_{2}$ , which together generate $T$ , and the subsemigroup generated by their setwise product $T_{1}T_{2}$ . In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$ . More generally, our technique merges semigroup homomorphisms from free semigroups.



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Author S. J. v. G. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant #655941. Author B. S. was supported by United States–Israel Binational Science Foundation #2012080 and NSA MSP #H98230-16-1-0047.



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[1] Almeida, J., Some algorithmic problems for pseudovarieties . Publ. Math. Debrecen 54(1999), no. suppl., 531552.
[2] Auinger, K. and Steinberg, B., On the extension problem for partial permutations . Proc. Amer. Math. Soc. 131(2003), no. 9, 26932703.
[3] Eilenberg, S., Automata, languages, and machines. Vol. B. Pure and Applied Mathematics, 59, Academic Press, New York, 1976.
[4] Gool, S. J. v. and Steinberg, B., Pointlike sets for varieties determined by groups. 2018. arxiv:1801.04638.
[5] Henckell, K., Pointlike sets: the finest aperiodic cover of a finite semigroup . J. Pure Appl. Algebra 55(1988), 85126.
[6] Henckell, K., Lazarus, S., and Rhodes, J., Prime decomposition theorem for arbitrary semigroups: general holonomy decomposition and synthesis theorem . J. Pure Appl. Algebra 55(1988), no. 1–2, 127172.
[7] Henckell, K., Rhodes, J., and Steinberg, B., Aperiodic pointlikes and beyond . Internat. J. Algebra Comput. 20(2010), no. 2, 287305.
[8] Krohn, K. and Rhodes, J., Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines . Trans. Amer. Math. Soc. 116(1965), 450464.
[9] Krohn, K., Rhodes, J., and Tilson, B., Algebraic theory of machines, languages, and semigroups. Academic Press, New York, 1968.
[10] Lee, E. W. H., Rhodes, J., and Steinberg, B., Join irreducible semigroups. 2017. arxiv:1702.03753.
[11] Place, T. and Zeitoun, M., Separating regular languages with first-order logic . Log. Methods Comput. Sci. 12(2016), Paper no. 5.
[12] Rhodes, J. and Steinberg, B., Pointlike sets, hyperdecidability and the identity problem for finite semigroups . Internat. J. Algebra Comput. 9(1999), no. 3–4, 475481.
[13] Rhodes, J. and Steinberg, B., The q-theory of finite semigroups. Springer Monographs in Mathematics, Springer, New York, 2009.
[14] Steinberg, B., A strange two-variable recursion. MathOverflow question, answered by M. Fischler.
[15] Steinberg, B., On pointlike sets and joins of pseudovarieties . Internat. J. Algebra Comput. 8(1998), no. 2, 203234.
[16] Straubing, H., Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science, Birkhäuser Boston Inc., Boston, MA, 1994.
[17] Wilke, T., Classifying discrete temporal properties . In: STACS’99, Lecture Notes. in Computer Science, 1563, Springer, 1999, pp. 3246.
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Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

  • Samuel J. van Gool (a1) and Benjamin Steinberg (a1)


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